CONTROL OF AN UNMANNED AERIAL VEHICLE ...

Proceeding of the 7th International Symposium on Mechatronics and its Applications (ISMA10), Sharjah, UAE, April 20-22, 2010

CONTROL OF AN UNMANNED AERIAL VEHICLE Hassan Noura

François Bateman

United Arab Emirates University Department of Electrical Engineering P.O. Box 17555, Al Ain, UAE [email protected]

Paul Cézanne University, Aix-Marseille III LSIS, UMR CNRS 6168 13397, Marseille, France [email protected]

ABSTRACT Unmanned Aerial Vehicles (UAV) have been attractive for years for many researchers and companies worldwide due to their commercial and military applications. The main problem of UAV is the absence of certification and the poor reliability that prevent their integration in the civil airspace. The problem can be overcome by improving the UAV nominal control and by designing a Fault-Tolerant Control System (FTCS). This paper deals with the design of a nominal control law of a UAV. Actually, planes are naturally unstable, that is why a particular attention must be paid in the design of a control law able to stabilize the system and let it track a predefined path. The design of a control law using eigenstructure assignment technique is developed in this paper and validated in simulation to a model of a small UAV. 1.

understand the flight mechanisms of the plane will be described. Then the model of the UAV used for this study built using Matlab/Simulink is presented. The study of the behavior of this UAV shows that the feedback control will highly improve its dynamical performances. Experiments are carried out to identify physical parameters useful for the model of the UAV. 2.

SYSTEM DESCRIPTION

The UAV considered in this work is a single-engined aircraft (Figure 1).

INTRODUCTION

Unmanned Aerial Vehicles (UAV) or aircrafts without pilot are becoming more and more popular for their military and civil applications. The main advantage of these systems is that it is very often possible to adapt the shape and the characteristics of these machines to their functions, thanks to the absence of pilot. Another advantage of these systems is that they allow achieving dangerous missions without putting operators at risk. The UAVs are generally integrated in generic systems including operators, ground stations, communication links, connections and various sensors. Several studies have shown the importance of designing control laws for aircrafts. The main objectives are to maintain the stability of the aircraft and to improve its maneuverability [1]. Several techniques have been designed for the control of a civil aircraft using Linear Quadratic Control, H∞, Model Predictive control, Fuzzy Logic, etc. Other robustness analyses are described in [2] and [3]. Another work discussed the development and testing of a unique benchmark consisting of a fleet of eight autonomous unmanned aerial vehicles [4]. This benchmark is used to compare several control approaches. In this paper, a model of a UAV is presented and a nominal control law is designed based on eigenstructure assignment technique. After a short presentation of the autonomous flight systems and components of the autopilot, a theoretical study to

Figure 1. The UAV. Some assumptions are made to simplify the model description of this UAV: The Earth is motionless: that means that the Coriolis acceleration is neglected and thus the reference frame related to the Earth can be considered as inertial Galilean. The Earth is flat: under this assumption, the centripetal acceleration can be neglected compared to gravity. Gravity does not vary according to altitude: the flight envelope of the drone should not exceed 100m altitude, which reinforces the fact that the variation of gravity on this field will be extremely weak. The total mass is constant: the fuel consumption of the engine of the drone being low, so the mass is considered as constant during the flight. The UAV is rigid-body: The plane being small (Length: 1.3 m, Wingspan: 1.60 m, Weight: 2.6 kg). The UAV is symmetrical: The aircraft has a symmetry plane which passes by its longitudinal axis. Also, the distribution of the masses is the same on both sides of the aircraft. Density, temperature and air pressure are supposed to be constant: The flight envelope of the UAV being limited in altitude to about 100m, the following values

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then will be considered: ρ0 = 1.225 kg/m3, T0 = 288 K, p0 = 101325 Pa. 2.1. Reference Frames of Flight Mechanics

Another rotation matrix Mba is used here from the aircraft reference frame Rb towards the aerodynamic reference frame Ra according to the angle of incidence α, and the angle of sideslip β as shown in Figure 3.  cos α cos β sin β sin α cos β    M ba =  − cos α sin β cos β − sin α sin β   − sin α  0 cos α  

Three reference frames are defined: the terrestrial reference frame (R0), the reference frame related to the UAV (Rb), and the aerodynamic reference frame (Ra).

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r r r The terrestrial reference frame R0 (G, x0 , y0 , z 0 ) : It is direct, supposed Galilean and originates in the center of gravity (c.g.) of the drone G, z is directed towards the center of the Earth and x axis is directed towards the True North. r r r The aerodynamic reference frame Ra (G , xa , ya , za ) : The origin is considered on the center of gravity (c.g.) G of the aircraft. The true air speed V is along with x axis, z axis is perpendicular to x and in the symmetry plane of the aircraft. r r r The plane reference frame Rb (G, xb , yb , zb ) : The origin is on the c.g. of the aircraft (b stands for body). The x axis is according to the fuselage centerline of the aircraft forwards. The y axis allows defining the pitch axis, and z the yaw axis. The transformation from the terrestrial reference frame R0 to the UAV reference frame Rb is obtained according to three rotations defined by the angles of Cardan (φ, θ, ψ) as shown in Figure 2 [5]. The rotation matrix Mb0 from the aircraft towards the terrestrial reference frame is defined by:  cos θ cos ψ sin ϕ sin θ cos ψ − cos ϕ sin ψ  M b 0 =  cos θ sin ψ sin ϕ sin θ sin ψ + cos ϕ cos ψ  − sin θ sin ϕ cos θ  cos ϕ sin θ cos ψ + sin ϕ sin ψ   cos ϕ sin θ sin ψ − sin ϕ cos ψ   cos ϕ cos θ 

Figure 3. Rotation from Ra to Rb.

3.

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UAV MODELLING

The UAV is considered as a platform with 6 degrees of freedom: 3 translations and 3 rotations. Based on the flight dynamics equations, the model of the UAV, illustrated by Figure 4, can be described by the following nonlinear state-space representation:

(3)

δx indicates the position of the throttle lever

δe is the elevator command

δa is the aileron command

δr is the rudder command

Figure 2. Rotation from Rb to R0. Figure 4. The UAV model.

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Through forward and backward movements, a quasi null plate is obtained. Then, the c.g. position can be read on the tool.

3.1. Experiments The complete model of this UAV is simulated to be as close as possible to the real UAV. For this reason, physical parameters and the physical limitations of the control surfaces are considered. These limitations are very important mainly in closed-loop where the calculated control laws may exceed these limitations. Since the laboratory is not equipped with appropriate tools and equipment, basic experiments are conducted to determine these parameters [6]. The method DATCOM can also be used to determine the model of aerodynamic coefficients [7]. The known parameters required to build the model are: Wingspan Aerodynamic Chord Profile of the wing Surface Length of the fuselage Horizontal stabilizer Mass Diameter of the propeller Engine

1.60m 0.28m Clark Y 0.448m ² 1.24m 0.55m 2.5kg 0.26m 7.5cc

Figure 6. Locating the center of gravity.

The physical limitations are measured when actuating the control surfaces using the remote control of the UAV as shown in Figure 5.

As for the y axis, since the aircraft has a longitudinal axis of symmetry, the c.g. is on the longitudinal axis. Regarding the z axis, the sides of the plane are maintained by two pivots which will be moved towards positive or negative z. When the plane does not tend to fall on the right-hand side or the left-hand side, the position of the c.g. is determined. Xcg = 75mm from the leading edge of the wing. Ycg = 0. Zcg = 60mm with the top of the fuselage centerline.

Coefficient of inertia of the UAV The coefficients of inertia are important to define the geometrical characteristics of the UAV. As said earlier, since no appropriate equipments are available, basic experiments, based on the pendulum theory, are conducted as shown in Figure 7.

Figure 5. Measuring the physical limitations. The Physical limitations are summarized in the following table. Min

δx 0

δa -7°

δe -16°

δr -16°

Max

1

7°

16°

16°

Table 1. Physical limitations of the control surfaces.

Center of gravity of the UAV

Figure 7. Swinging of the plane as a simple pendulum.

The determination of the center of gravity (c.g.) of the UAV is very important since it is the origin of the plane references and it allows to know if the aircraft is stable in open-loop. A tool provided with this UAV facilitates the experiment to locate the center of gravity. The wings of the UAV lay on the tool as shown in Figure 6 and one has to observe the plate of the aircraft.

The advantage of this experiment is to be simple. It consists of swinging the plane thanks to a carrying framework itself suspended. To determine the coefficients of inertia around an axis, it is enough to place this axis parallel to the axis of the oscillations. One can thus determine the coefficients of inertia

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around axis x, y, z, and of a fourth axis belonging to the plan (x, O, z), the axis xz. Direct calculations give the coefficients of inertia of the UAV + frame. Two experiments will be conducted: the frame alone and the UAV + frame. The final results are then obtained by subtraction. These experiments helped in the setup of the model of the aircraft. This nonlinear model is then implemented in Matlab/Simulink to simulate the total behavior of the UAV. The objective is to design a nominal control law (an autopilot) allowing the UAV to follow a predefined track.

4.

Design of the control law

Among all possible control laws that may be applied to the UAV, the eigenstructure assignment technique is selected to be used in this application. The rationale to use this technique is that when studying fault-tolerant control later, it would be possible to decouple some fault effects on safe actuator. To design this control law, the linearization is carried out in two stages and leads to two models: a longitudinal model and a lateral model. Also, it has to be noticed that at this stage, the trajectory of the UAV is not controlled.

3.2. Linearization The design of a linear control law is carried out using the linearized model of the UAV around an operating point. The linearized model can be written under the following state-space representation:  x& = A x + B u (4)  y = C x Where x ∈ R12 , u ∈ R 4 , and y ∈ R12 are the state vector, the control vector, and the output vector, respectively. For a level flight, a constant altitude at 200m and a constant speed of 22m/s are considered. 0 0 0 0 0 1 0 0.14 0 0 0  0   0 0 0 0 0 0 0 1 0 0 0 0   0 0 0 0 0 0 0 0 1.01 0 0 0   0 − 9 . 81 0 − 0 . 06 9 . 75 0 0 0 0 0 0 0   0 0 0 − 0.04 − 2.48 0 0 0.98 0 0 0 0   0 0 0 0 − 0.38 0.14 0 − 1 0 0 0  0.44 A=  0 0 0 0 − 31.93 − 4.73 0 2.28 0 0 0  0  0 0 0 0 − 88.05 0 0 − 7.81 0 0 0 0   0 0 0 0 6.42 − 0.62 0 − 0.25 0 0 0  0  0 − 0.08 0 1 − 0.08 0 0 0 0 0 0 0   0 22 0 0 0 0 0 0 − 0.08 0 22 0   − 22 0 0 22 0 0 0 0 0 0 0  0

 0   0  0   1.33 − 0.01   0 B=  0  0   0  0   0  0 

0   0  0 0 0   0 0.06 0  0 − 0.03 0   − 0.03 0 0.08   − 43.1 0 − 1.2  0.67 − 31.87 0   − 1.67 0 − 8.29 0 0 0   0 0 0  0 0 0  0 0

4.1. Longitudinal Control Law The objective here is to control the speed (V) and the altitude (Z) of the UAV. A level flight is considered in this study. For this purpose, the control inputs are the throttle (δx) and the command of the elevator (δe). The longitudinal model can then be written as:

x& L = AL x L + B L u L  yL = CL xL

with x L = [θ v α q Z 0 ] , u L = [δx δe ]T , and y L = [v Z 0 ] . T

T

A tracking control law based on a state-feedback with integrator is setup:

u L = −K L xL − LL z L

(6)

Where zL is the integrator of the error vector: eL = yr – yL. yr is the reference vector that the output yL should track.

∫

z L = ( y r − y L ) dt

(7)

Considering the augmented system where the augmented state vector is given by X L = [x L z L ]T , leads to: T  X& = A Lon X L + B Lon u L + [0 I ] y r  L  y L = [C L 0] X L

0 0

Matrix C is considered to be the identity matrix.

(5)

(8)

Therefore, the state feedback control law uL defined by (6) can be given by:

u L = − K Lon X L = −[K L LL ]X L The state-feedback gain KLon is then eigenstructure assignment technique:

( ALon − B Lon K Lon )vi = λi vi v [ ALon − λi I BLon ]  i  = 0 wi 

(9) calculated using

(10)

where λi and vi are respectively eigenvalues and eigenvectors of ALon - BLon.KLon.

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Setting the eigenvalues of the closed-loop system λi and solving (10) provide two matrices VLon = [v1 … v7] and WLon = [w1 … w7]. Then, the state feedback gain KLon will be given as: −1 K Lon = [K L L L ] = W Lon V Lon

A level flight is considered in this first study. Figure 9 shows the performance of this control law for an increase in the reference of the speed of the UAV (Vr) from 22 m/s to 23 m/s.

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The tuning of the eigenvalues aims at controlling the time response and the damping ratio of the second order modes and the time response of the first order modes of the UAV.

4.2. Lateral Control Law Here, the objective is to control the roll angle ϕ and the yaw rate β. In this case, the control inputs are the commands of the aileron (δa) and the rudder (δr). The lateral model can be given by the following state-space representation:

x& l = Al xl + Bl ul   y l = Cl x l

Figure 9. The response of the speed of the UAV. In response to the change of the reference, the main control input, which is the throttle (δx) is displayed in Figure 10.

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with xl = [ϕ β p r ]T , u L = [δa δr ]T , and y L = [ϕ β ]T . A tracking lateral control law is then designed in a way similar to the longitudinal controller. The state-feedback control law is given by:

ul = −K Lat X l = −[K l Ll ]X l

Figure 10. The command of the throttle.

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KLat is calculated by setting the eigenvalues of the closed-loop lateral system. Therefore, the longitudinal and the lateral control laws are applied to the UAV as described by Figure 8.

A change in the reference of the altitude from 200 m to 203 m and then to 197 m is also tested and results are shown on Figure 11. Figure 12 shows the reaction of the command of the throttle in response to the step changes in the altitude reference.

Figure 11. The response of the altitude of the UAV.

Figure 8. Longitudinal and Lateral Control Law. Variables (.)e denotes the operating point values of these variables which should be considered in the simulation.

5.

RESULTS

Figure 12. The command of the throttle.

This control law is implemented in Matlab/Simulink environment. The model of the UAV is implemented as an Sfunction.

Figure 13 shows a step change in the reference of the roll angle ϕ. Control inputs (δa and δr) in response to this step change are illustrated by Figures 14 and 15.

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7.

[1] Magni J. F., Bennami S., and Terlouw J., Robust Flight Control, a design challenge. Springer, 1997. [2] Duc G., Commande H∞ et µ-Analysis, des outils pour la robustesse. Hermes, 1999 [3] Zhou K., Doyle J.C., and Glover K., Robust and Optimal Control. Prentice Hall, 1996. [4] How J., King E., and Kuwata Y., "Flight Demonstrations of Cooperative Control for UAV Teams," AIAA 3rd "Unmanned Unlimited" Technical Conference, September 2004, Chicago, Illinois. [5] Bateman F., "Diagnostic actif et tolérance aux défauts majeurs d’actionneurs : application à un drone," PhD Thesis, University Paul Cezanne, Aix-Marseille III, 2008. [6] Miller M.P. "An accurate method for measuring the moments of inertia of airplanes," Technical Notes National Advisory Committee for Aeronautics. [7] Raymer D. P., Aircraft design, a conceptual approach. AIAA Education series, 1998.

Figure 13. The response of the roll angle.

Figure 14. The command of the aileron.

Figure 15. The command of the rudder. These results show the efficiency of the longitudinal and lateral control laws using eigenstructure assignment designed and applied to this UAV.

6.

REFERENCES

CONCLUSIONS

In this paper, a modeling of a small UAV is presented. The obtained nonlinear model is then linearized around an operating point. Two models are described: a longitudinal model and a lateral one. For each of them, a linear control law is designed. These control laws are setup using the eigenstructure assignment technique. The results obtained in this study are encouraging. However, several improvements are required because the modeling uncertainties are not taken into account. These uncertainties exist due to the basic techniques used in the modeling process. The robustness to these uncertainties will be considered in future works. The implementation of such techniques to the real UAV using a GPS, an attitude sensor MTI, and a microcontroller, is ongoing. The final objective is to develop fault diagnosis and fault-tolerant control techniques applied to UAVs.

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