D A Mercado
email: dmercado@hds.utc.fr

P Castillo
email: pedro.castillo@hds.utc.fr

R Lozano
email: rlozano@hds.utc.fr

Quadrotors Data Fusion using a Particle Filter*

This paper presents a data fusion algorithm by means of a particle filter (PF), for estimate the position of a quadrotor equipped with multiple sensors. A global positioning system (GPS) is considered for position measurement in a loosely coupled scheme, velocity can be obtained from an optic flow sensor, whilst an inertial measurement unit (IMU) can provide orientation and angular rate measurements. Simulations were carried out, where real noise from a not expensive GPS is added to the simulated position to test the proposed algorithm.

I. INTRODUCTION

Unmanned Aerial Vehicles (UAVs) have gained great relevance in the last years thanks to their huge potential in many civilian and military applications such as exploration, surveillance, search and rescue, in a cheap way and without risking human lives in dangerous situations. Particularly, four rotors rotorcrafts, also known as quadrotors, have received special interest since their rotor's configuration produces cancellation of the reactive torques, considerably simplifying their analysis and control. Also, they are suitable for vertical take off and landing, as well as hovering, making them a good choice for maneuvering in small spaces or perform high precision tasks. Several stabilization and trajectory tracking control laws have been proposed for this kind of systems and validated through simulations and experiments in indoor applications, relying on a good measurement of the position and velocity. To cite some examples, [START_REF] Castillo | Modelling and control of mini-flying machines[END_REF] proposes an attitude stabilization control strategy for hover flight using nested saturations, while in [START_REF] Xu | Sliding mode control of a quadrotor helicopter[END_REF] a sliding mode approach is used to accomplish position control of a quadrotor. In [START_REF] Guisser | Stabilization and trajectory tracking in discrete-time of an autonomous four rotor mini-rotorcraft[END_REF] a trajectory tracking control by means of a discrete time feedback linearization control scheme is proposed. However, big effort is still required to accomplish autonomous flight for outdoor applications, due to presence of external perturbations, especially the wind, and the lack of a good measurement for the position and velocity. Not expensive Global Positioning Systems (GPS) sensors can provide an estimation of the position and velocity, however, the errors, of 2m at best, and their low measurement rate of about 5Hz, are not suitable for precise applications and can interfere with the system stability, even more, GPS can easily lost its signal leaving the system without a position measurement. Another alternative widely studied are the optical flow sensors which use computer vision algorithms for estimating the motion velocity of a system, but they are noisy and sensibles for lighting changes. Data fusion algorithms are an interesting solution to this problem, they take information from multiple sensors, especially GPS, cameras and IMU, to improve the estimation of the position and velocity. The Kalman filter an its variations are the mos popular approach, for example, in [5] an observer-control scheme for quadrotors using the Extended Kalman Filter (EKF) is proposed and tested in real time indoor experiments, using data from an optic flow algorithm and an IMU. In [6], a Robust Adaptive Kalman Filter is proposed for estimation of UAV dynamics in the presence of sensor faults (see also [7], [8], [START_REF] Wang | Integration of GPS/INS/vision sensors to navigate unmanned aerial vehicles[END_REF]). However, this techniques assume that the process and measurement noise are Gaussian, which is not the case of a GPS in a loosely coupled scheme. The great advances in micro controllers processing capacity allow to explore other alternatives. The Particle Filter (PF) presents an interesting option to this problems, at the cost of computational expense, because it is not restricted to Gaussian noise. In ?? is presented a vision based algorithm using a particle filter tracking for estimating the position of a quadrotor. In this paper the use of the PF is explored, through simulations, for data fusion and state estimation in quadrotors equipped with multiple sensors in outdoor missions. Particularly, a GPS is considered for position measurement. Real GPS data was collected and the obtained error was added to the simulated position. The paper proceeds as follows. In section II the dynamic model of the quadrotor is given. In section III, the PF for the quadrotor state estimation is proposed. In section IV the performance of the PF is tested in simulation. Section V gives the conclusions and future work.

II. QUADROTOR DYNAMIC MODEL

The quadrotor can be represented as a rigid body in space with mass m and inertia matrix J, subject to gravitational and aerodynamic forces. Let us consider an inertial coordinate frame I = {X Y Z}, fixed to the ground and a body fixed coordinate frame, B = {e 1 , e 2 , e 3 } (see Fig. 1). Consider the vectors

ξ = [x y z] T (1)
Φ = [φ θ ψ] T (2)
which stand for the position of the center of gravity, with respect to the inertial frame I, and the Euler angles roll, pitch and yaw, respectively. The motion equations are given by the Newton-Euler equations in the inertial frame I [10]

m ξ = T Re 3 -mge 3 (3) J Ω = -Ω x JΩ + Γ (4)
where T ∈ ℜ + is the total thrust of the motors, g is the gravity constant and Γ ∈ ℜ 3 is the input torque defined in the body fixed frame B. R ∈ SO(3) : B → I is the rotational matrix from the body frame to the inertial frame.

Ω = [p q r] T (5)
represents the angular velocity in the body frame B. Ω x stands for the skew symmetric matrix such that Ω x υ = Ωxυ is the vector cross product. The kinematic relation between the generalized velocities Φ = ( φ , θ , ψ) and the angular velocity Ω is given by [START_REF] Roskam | Airplane Flight Dynamics and Automatic Flight Controls, Part I[END_REF]]

Ω = Q Φ (6) with Q =   1 0 -sθ 0 cφ cθ sφ 0 -sφ cθ cφ   (7) III. PARTICLE FILTER Consider the state vector χ ∈ ℜ 12 χ = [ξ ξ Φ Φ] T (8)
Then from ( 3), (4), with a sampling time T s small enough, the discrete time evolution model is given by

χk =     ξk-1 T s + ξ k-1 ( 1 m T k-1 R k-1 e 3 -ge 3 )T s + ξk-1 Φk-1 T s + Φ k-1 Γk-1 T s + Φk-1     + ω k (9)
where the total thrust T and the control torque Γ are the system inputs. ω stands for the model noise with a probability density p ω . Consider also the measurement vector Y = [ξ GPS ξOF Φ IMU ΦIMU ] T , it is the position measured by a GPS, the velocity measured by an optic flow sensor and the orientation and angular rate obtained by an IMU. Hence, the observation model is

Y k = χ k + ν k (10)
ν represents the measurement noise with a probability density p ν . The particle filter (PF) or sequential Monte Carlo technique is a kind of recursive Bayesian filter for state estimation [START_REF] Chen | Bayesian filtering: From kalman filters to particle filters, and beyond[END_REF]. It consists in generating N p particles according to the a priori probability, with an importance weight W defined as

W k = p(Y 1:k |χ 0:k )p(χ 0:k ) q(χ 0:k |Y 1:k ) (11)
where p(Y |χ) is the probability density function of Y given χ, and q(χ k |Y k ) is the so-called proposal distribution.

At each iteration, the outputs are measured by the sensors and the weights can be updated recursively, using the bootstrap method, by the following expression

W i k = W i k-1 p(Y k |χ i k ) = W i k-1 p ν (Y k -χ i k ) i = 1, ..., N p (12)
for a normal probability distribution in the measurement, with covariance matrix

Q W i k = W i k-1 e (-1 2 (Y k -χ i k )Q -1 (Y k -χ i k )) (13)
normalizing

W i k = W i k Σ N p i=1 W i k (14)
Then, the set of particles evolves following the evolution model ( 9), with ω i k randomly generated according to p ω . In order to avoid the weight degeneracy problem, a resampling step is required. In order to do so, the effective sample size N e f f is employed as a measure of the number of active particles

N e f f = 1 Σ N p i=1 (W i k ) 2 (15)
it can be noticed that this value is maximum (N e f f = N p ) when all the particles have the same weight, and minimum (N e f f = 1) when only one weight is different from zero. If the effective sample size is lower than a certain threshold N e f f < N th , the resampling step is performed by means of the Kitagawa method [START_REF] Kitagawa | Monte Carlo filter and smoother for non-Gaussian nonlinear state space models[END_REF].

The estimated stated is given by the particles barycenter, it is

χk = Σ N p i=1 W i k χ i k (16)
Since GPS sample frequency is slow (5Hz) with respect to the others sensors, and in order to deal with the GPS loss signal, the measured position is updated in the following form

ξ GPSk = ξ GPSk-n + nT s ξk ( 17
)
where n is the number of sample times T s passed since the last valid GPS data.

IV. SIMULATIONS Simulations were carried out to test the proposed estimation algorithm for a quadrotor with an inaccurate position measurement, which is the case for a not expensive GPS. GPS errors were obtained from real experiments in an urban environment surrounded by buildings, and added to the simulated actual position, taking also into account the GPS measurement rate (5Hz) and including signal losses of 20 seconds every 100 seconds. The described GPS errors can be depicted in Fig. 2,where the zero values every 100 seconds represent signal losses. Optical flow can provide a noisy velocity measurement, hence some with noise was added to the velocity signal. The noisy measured velocity is shown in Fig. 3. The performance of the particle filter for position estimation while the UAV evolves in a spiral trajectory can be seen in Fig. 4, using N p = 100 particles and a resampling threshold of N th = 10. It can be noticed that the PF performs an excellent position estimation despite the poor position measurement, this is clearer from Fig. 5 where the estimation error is displayed, which can be compared against the measurement error in Fig. 2. It is clear that the position estimation greatly improved the measured one and perfectly handled the signal losses and from measurement errors of until 20 meters are obtained error of less than one meter, making it suitable for precision tasks. In order to provide a comparison against other estimation techniques, the popular EKF was also tested under the same conditions and the obtained estimation errors are presented at Fig. 6, where it can be seen that the errors are much higher than the obtained with the PF, this can be explained due to the fact that the EKF assumes that measurement noise is Gaussian, which is not the case for a GPS in a loosely coupled scheme (see Fig. 2). Finally, Figs. 7 and 8 contain the XY plane view and the three dimensional one with the real and estimated positions.

V. CONCLUSIONS AND FUTURE WORK

Simulations have shown promising results of the proposed data fusion strategy, considerably reducing the measurement error while handling the GPS signal losses. The PF proved to be more adequate in this case than the EKF, since the measurement noise of a GPS in a loosely coupled scheme is not Gaussian. Future work includes to implement the proposed strategy embedded on a quadrotor equipped with a GPS sensor, for real time experiments. It is also desired to use the estimated data to close the loop in a control strategy to accomplish trajectory tracking for outdoor applications.

Fig. 1 .

 1 Fig. 1. Quadrotor in an inertial reference frame.

Fig. 2 .Fig. 3 .

 23 Fig. 2. GPS measurement error.

Fig. 5 .

 5 Fig. 4. Position.

Fig. 7 .

 7 Fig. 6. EKF estimation error.

LAFMIA UMI

CINVESTAV-CNRS, Mexico.

*This work was supported by project SEARCH


Diego Mercado
email: dmercado@utc.fr

Pedro Castillo
email: castillo@hds.utc.fr

R Castro
email: rcastro@cinvestav.mx

Rogelio Lozano
email: rlozano@utc.fr

2-Sliding Mode Trajectory Tracking Control and EKF Estimation for Quadrotors

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Unmanned Aerial Vehicles (UAVs) have gained great relevance in the last years thanks to their huge potential in many civilian and military applications such as exploration, surveillance, search and rescue, in a cheap way and without risking human lives in dangerous situations. Particularly, four rotors rotorcrafts, also known as quadrotors, have received special interest since their rotor's configuration produces cancellation of the reactive torques, considerably simplifying their analysis and control. Also, they are suitable for vertical take off and landing, as well as hovering, making them a good choice for maneuvering in small spaces or perform high precision tasks. Several stabilization and trajectory tracking control laws have been proposed for this kind of systems and validated through simulations and experiments in indoor applications, relying on a good measurement of the position and velocity. To cite some examples, [START_REF] Castillo | Modelling and control of mini-flying machines[END_REF] proposes an attitude stabilization control strategy for hover flight using nested saturations, while in [START_REF] Xu | Sliding mode control of a quadrotor helicopter[END_REF] a sliding mode approach is used to accomplish position control of a quadrotor. In [START_REF] Guisser | Stabilization and trajectory tracking in discrete-time of an autonomous four rotor mini-rotorcraft[END_REF] a trajectory tracking control by means of a discrete time feedback linearization control scheme is proposed. A second order sliding mode algorithm for attitude control is achieved in [START_REF] Derafa | Super Twisting Control Algorithm for the Four Rotors Helicopter Attitude Tracking Problem[END_REF]. However, big effort is still required to accomplish autonomous flight for outdoor applications, due to presence of external perturbations, especially the wind, and the lack This work was supported by CONACYT and project SEARCH. of a good measurement for the position and velocity. Not expensive Global Positioning Systems (GPS) sensors can provide an estimation of the position and velocity, however, the errors, of 2m at best, and their low measurement rate of about 5Hz, are not suitable for precise applications and can interfere with the system stability, even more, GPS can easily lost its signal leaving the system without a position measurement. Another alternative widely studied are the optical flow sensors which use computer vision algorithms for estimating the motion velocity of a system, but they are noisy and sensibles for lighting changes. Data fusion algorithms are an interesting solution to this problem, they take information from multiple sensors, especially GPS, cameras and IMU, to improve the estimation of the position and velocity, for example, in [START_REF] Munoz | Observercontrol scheme for autonomous navigation: Flight tests validation in a quadrotor vehicle[END_REF] an observercontrol scheme for quadrotors using the EKF is proposed and tested in real time indoor experiments, using data from an optic flow algorithm and an IMU (see also [START_REF] George | Tightly coupled INS/GPS with bias estimation for UAV applications[END_REF], [START_REF] Ding | Adaptive filter design for UAV navigation with GPS/INS/optical flow integration[END_REF], [START_REF] Wang | Integration of GPS/INS/vision sensors to navigate unmanned aerial vehicles[END_REF]). In this work, it is considered the problem where a quadrotor is used in an exploration outdoor flight mission, where it has to track a desired position under difficult conditions, where the measurement error is big and external perturbations affect the system. An study is presented through simulations, about the effect of such errors on the controlled system behavior. For the control strategy, a time scale separation along with a second order sliding mode control for the translational dynamics and a linear PD controller for the rotational dynamics are used to accomplish the trajectory tracking. In order to improve the performance of the control scheme under bad mea-Fig. 1. Quadrotor in an inertial reference frame.

surements, an estimation algorithm is developed by using the EKF to fuse the data from multiple commonly used sensors, such as GPS, vision systems and IMU to measure position, velocity and orientation, respectively.

QUADROTOR DYNAMIC MODEL

The quadrotor can be represented as a rigid body in space with mass m and inertia matrix J, subject to gravitational and aerodynamic forces. Let us consider an inertial coordinate frame I = {X Y Z}, fixed to the ground and a body fixed coordinate frame, B = {e 1 , e 2 , e 3 } (see Fig. 1). Consider the vectors

ξ = [x y z] T (1)
Φ = [φ θ ψ] T (2)
which stand for the position of the center of gravity, with respect to the inertial frame I, and the Euler angles roll, pitch and yaw, respectively. The motion equations are given by the Newton-Euler equations in the inertial frame

I [9] m ξ = T Re 3 -mge 3 (3)
J Ω = -Ω x JΩ + Γ (4)
where T ∈ + is the total thrust of the motors, g is the gravity constant and Γ ∈ 3 is the control torque defined in the body fixed frame B. R ∈ SO(3) : B → I is the rotational matrix from the body frame to the inertial frame. Ω = [p q r] T represents the angular velocity in the body frame B. Ω x stands for the skew symmetric matrix such that Ω x υ = Ωxυ is the vector cross product. The kinematic relation between the generalized velocities Φ = ( φ, θ, ψ) and the angular velocity Ω is given by [10]

Ω = 1 0 -sθ 0 cφ cθsφ 0 -sφ cθcφ Φ (5)

CONTROL STRATEGY

Assuming that the closed loop dynamics of rotation is faster than the translational one, it is possible to separate the model in two independent subsystems [START_REF] Bertrand | A hierachical controller for miniature VTOL UAVs: Desing and stability analysis using singular perturbation theory[END_REF]. The strategy consists on designing a controller for the translational dynamics such that it guarantees the trajectory tracking by providing the desired orientation to an inner control loop. Due to its robustness property and in order to attenuate the chattering effect, a second order sliding mode control have been chosen [START_REF] Shtessel | S liding Mode Control and Observation[END_REF] to deal with uncertainties and perturbations in the translational dynamics, while tracking a desired position. The attitude is controlled by a linear PD controller. Finally, in the next section, an estimation algorithm by means of an EKF is presented, it fuses information taken from multiple sensors to improve the state estimate that is used for the trajectory tracking and attitude stabilization controllers. The whole control scheme is presented in the block diagram at Fig. 2.

Second Order Sliding Mode Trajectory Tracking Control

Defining the desired position ξ d and the position error ξ = ξ -ξ d , and substituting into (3) leads to

m ξ = (T Re 3 ) d -mge 3 -m ξd + w (6)
where w ∈ 3 is an external disturbance vector. Let us consider the so called switching function with relative degree 2

σ = ξ + k ξ dt (7)
where k ∈ 3x3 is a diagonal positive definite gain matrix.

Then calculating the second time derivative

σ = ξ + k ξ = (T Re 3 ) d -mge 3 -m ξd + k ξ (8)
Consider now u = (T Re 3 ) d to be the control input. Then using the Twisting Algorithm [START_REF] Shtessel | S liding Mode Control and Observation[END_REF], the following discontinous controller is proposed

u = m(ge 3 + ξd -k 1 ξ -r 1 Sgn(σ) -r 2 Sgn( σ)) (9)
with

Sgn(x) = sgn(x 1 ) sgn(x 2 ) sgn(x 3 ) ; sgn(x) = 1 x > 0 -1 x < 0 ( 10
)
and r 1 , r 2 ∈ are constant control parameters.

The closed loop dynamics is obtained substituting (9) in [START_REF] Wang | Integration of GPS/INS/vision sensors to navigate unmanned aerial vehicles[END_REF], it is

σ = -r 1 Sgn(σ) -r 2 Sgn( σ) + 1 m w (11)
In order to analyze the stability of the closed loop system, let us consider the Lyapunov candidate function

V = r 1 ||σ|| 1 + 1 2 σT σ ( 12
)
which is Lipschitz continuous. Differentiating [START_REF] Shtessel | S liding Mode Control and Observation[END_REF] with respect to time leads to V = r 1 Sgn(σ) T σ + σT σ [START_REF] Shevitz | Lyapunov stability theory of nonsmooth systems[END_REF] everywhere but on σ = 0 where V is not differentiable. By substituting the closed loop dynamics [START_REF] Bertrand | A hierachical controller for miniature VTOL UAVs: Desing and stability analysis using singular perturbation theory[END_REF] in the above

V = r 1 Sgn(σ) T σ + (-r 1 Sgn(σ) -r 2 Sgn( σ) + 1 m w) T σ(14)
or, equivalently,

V = -r 2 || σ|| 1 + 1 m w T σ ( 15
)
and thus

V ≤ || σ|| 1 (-r 2 + 1 m ||w||) (16)
if the external disturbance is bounded ||w|| ≤ ma, for some a > 0, and choosing r 2 > a leads to V ≤ 0 (17

)
The set S where V = 0 is given by

S = {(σ, σ)| σ = 0} (18)
on this set, the trajectories of the system take the form

σ = -r 1 Sgn(σ) -r 2 Sgn(0) + 1 m w (19)
where

Sgn(0) ∈ [-1, 1]. An invariant set on S implies σ = 0, hence 0 = -r 1 Sgn(σ) -r 2 Sgn(0) + 1 m w (20)
selecting r 1 > r 2 + a is enough to assure that the largest invariant set on S contains only the origin σ = σ = 0. Using the extended version of the LaSalle invariance principle [START_REF] Shevitz | Lyapunov stability theory of nonsmooth systems[END_REF], all the trajectories of the system converge to the origin, and the system is stable. A formal proof for globally equiuniformly finite time stability for this kind of systems with the proposed gains conditions can be found in [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

So the sliding mode control law (9) solves the trajectory tracking problem despite uncertainties and perturbations.

It is important to notice that

R d e 3 = R dx R dy R dz = (T Re 3 ) d T d (21)
with T d = (T Re 3 ) d . Hence, with a constant ψ d , it is possible to write φ d and θ d explicitly as

φ d = arcsin - R dy -R dx tan(ψ d ) sin(ψ d )tan(ψ d ) + cos(ψ d ) (22) θ d = arcsin R dx -sin(φ d )sin(ψ d ) cos(φ d )cos(ψ d ) (23)
For the attitude stabilization control, a proportionalderivative controller is proposed, that acts on the orientation error defined by Φ = Φ -Φ d , this is

Γ = -k do Φ -k po Φ (24)
with the gain matrices k do , k po ∈ 3x3 are diagonal positive definite.

A similar control strategy have already been tested in real time indoor experiments [START_REF] Mercado-Ravell | Quadrotors Flight Formation Control Using a Leader-Follower Approach[END_REF], where a good position and velocity measurement were available (a video can be watched at: https://www.youtube.com/watch?v=b7A-WgegIuY). To really explode the great potential of UAV in exploration, surveillance, search and rescue applications, it is desired to extend the results to outdoor experiments. However, serious problems arrive due to the lack of a good measurement of the position and velocity of the UAV.

ESTIMATION ALGORITHM

This section deals with the issue of estimate the position and velocity of the quadrotor by means of a data fusion algorithm using an EKF, taking information from multiple sensors whose measurements are inaccurate and noisy.

Extended Kalman Filter

The EKF is a well known, powerful and widely used estimation algorithm for the case of non linear systems of the form

χ k+1 = f (χ k , u k ) + ω k (25) Z k = g(χ k ) + ν k ( 26
)
where χ is the state vector and Z is the measurement vector. The process and measurement noise ω, ν are supposed to be Gaussian with known covariance matrices Q and Ω, i.e.

ω ∼ N (0, Q) (27) ν ∼ N (0, Θ) (28)
Then, the a priori state estimates is given by χk+1|k = f ( χk|k , u k|k ) (29)

and the estimated outputs is

Y k+1|k = g(χ k+1|k ) ( 30
)
The a priory error covariance can be calculated as

P k+1|k = AP k|k A T + Q k (31)
where A = [ δf δχ ( χ, u)]. Now, the Kalman gain matrix can be determined

K k+1|k = P k+1|k G T (GP k+1|k G T + Θ k+1 ) -1 (32)
with G = [ δg δχ ( χ)]. After measuring the process, the a posteriori state estimate χ and error covariance

P k are respectively χk+1|k+1 = χk+1|k + K k+1 (Z k+1 -Y k+1|k ) (33)
P k+1|k+1 = (I -K k+1|k G k+1 )P k+1|k (34)

Model Adaptation

Consider the state vector

χ = [ξ ξ Φ Φ] T (35)
and the measurement vector

Z = [ξ GP S ξOF Φ IM U ΦIMU ] T (36)
it is, the position measured by a GPS, the velocity obtained by a vision system and an optic flow algorithm (an extra height sensor is required for the z coordinate) and the orientation and angular speed given by an IMU. The discrete time process dynamics, considering a sample time T s small enough is given by

χk+1 =      ξk T s + ξ k ( 1 m T k R k e 3 -ge 3 )T s + ξk Φk T s + Φ k Γ k T s + Φk      (37)
Since GPS sample frequency is slow (5Hz) with respect to the others sensors, and in order to deal with the GPS loss signal, the measured position is updated in the following form

ξ GP Sk = ξ GP Sk-n + nT s ξk ( 38
)
where n is the number of sample times T s passed since the last valid GPS data.

SIMULATIONS

Simulations were carried out to observe the behavior of the proposed observer-control scheme with external perturbations and inaccurate measurements from a GPS sensor. In order to introduce the errors from the GPS in a more realistic way, actual GPS data was collected under bad conditions for the GPS receiver, it is in an urban environment surrounded by buildings, then this real data was added to the simulated actual position of the quadrotor, considering also the rate of the GPS measures (5Hz) and including signal loss for twenty seconds every 100 seconds. Fig. 3 shows the described GPS measurement error, where the zero values every 100 seconds represent signal losses. A second sensor was considered to measure the velocity, with some white noise (Fig. 4 shows the noisy measured velocity), this can represent for example the measure from an optic flow sensor. Also, it was added an external perturbation ω like the one showed in Fig. 5 to test the robustness of the sliding mode controller. It is desired to follow a spiral trajectory in the XY plane, centered in the origin, at a constant height of 3 meters. The performance of the estimation algorithm can be studied through Fig. 6, where the estimation error is presented. It can be noticed that the estimator improve considerable the measured data and perfectly handles with the missing data while GPS signal is lost. Fig. 7 shows the orientation of the helicopter. Position and tracking errors are presented in Figs. 8 and9, showing good tracking results despite the measurement error and external perturbations. The control input u is shown in Fig. 10. Finally, Figs. 11 and 12 contain the real and desired position in the XY plane and in three dimensional space.

CONCLUSIONS AND FUTURE WORK

Simulations have shown promising results of the proposed control strategy for outdoor flight applications, despite the presence of perturbations and the errors introduced by sensors.

The presented estimation algorithm considerably improves the measured data and automatically deals with the loss data from sensors, helping to improve the performance of

Fig. 2 .

 2 Fig. 2. Control scheme block diagram.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. GPS measurement error.

Fig. 6 .Fig. 8 .

 68 Fig. 6. Estimation error.

Fig. 9 .Fig. 10 .Fig. 11 .

 91011 Fig. 9. Tracking error.

Fig. 12

 12 Fig. 12. 3d. the overall control scheme. Future work includes to implement the proposed strategy on a quadrotor equipped with a GPS sensor, for real time outdoor experiments.