2011 ECTC Proceedings ASME Early Career Technical Conference Hosted by ASME District E and University of Arkansas Support Provided by the ASME Old Guard and the Committee on Early Career Development March 31-April 02, 2011, Fayetteville, AR

DESIGN OF AN AUTOMATIC LANDING SYSTEM FOR A UAV USING FEEDBACK LINEARIZATION METHOD
Ghassan. Atmeh1 and Zeaid, Hasan2
1

Mechanical and Aerospace Engineering Department, The University of Texas at Arlington, Arlington, Texas, email: atmehg@gmail.com 2 Mechanical Engineering Department, Texas A&M University, College Station, Texas, email: zeadnws@hotmail.com CmδE Cn CY CYo CYβ CYδR CYδA c d D g J L L m q R = = = = = = = = = = = = = = = = = Pitching moment coefficient due to elevator deflection Yawing moment coefficient Side-force coefficient Side-force coefficient for zero angle of attack Side-force coefficient contribution due to side-slip angle Side-force coefficient contribution due to rudder deflection Side-force coefficient contribution due to aileron deflection Wing cord (m) UAV displacement from glide path (m) Drag fore (N) Acceleration due to gravity (m/s2) Inertia matrix Lift force (N) Applied moment vector (N.m) Mass (kg) Dynamic pressure Slant range (m)

ABSTRACT An automatic landing system for an unmanned aerial vehicle (UAV) is presented in the following paper. The nonlinear aircraft model with elevator deflection and thrust as control inputs is established using the appropriate aerodynamic data, followed by the definition of the flight trajectory the airplane is expected to travel during landing, which is divided into glide path and flare. Nonlinear control using feedback linearization method is employed to develop the automatic landing controller for the UAV aircraft. Elevator deflection is used to control the pitch angle and throttle controls the speed. The feedback linearization control technique provides reliable tracking performance with respect to a given landing trajectory. A nonlinear simulation is run using Matlab/Simulink to assess the controller integrity. Very well tracking is shown for glide path and velocity. An exponential trajectory is defined for flare, which needs a tighter control and hence requires both the altitude and sink rate for feedback purposes. The auto-landing system designed in this paper is meant to increase the autonomy of the UAV to eventually reach a fully autonomous system. NOMENCLATURE CD = Drag coefficient CDo = Drag coefficient for zero angle of attack CL = Lift coefficient CLo = Lift coefficient for zero angle of attack CLα = Lift coefficient contribution due to angle of attack CLδE = Lift coefficient contribution due to elevator deflection Cl = Rolling moment coefficient Cm = Pitching moment coefficient Cmα = Pitching moment coefficient due to angle of attack

INTRODUCTION The landing stage of flight is recognized as a difficult challenge for autonomous controller design. Auto-landing systems designed for aircrafts must take into account external disturbances, such as wind and turbulence, as well as uncertainties in the plant model used for the design of the controller. In addition, the auto-land system must consistently control an aircraft to an accurate touchdown point and along with smooth touchdown to prevent damage to the aircraft. As indicated in Ref [1]. The landing phase accounts for 60% of general aviation accidents, hence, it is mandatory to have the ability to land an airplane safely, especially while having to deal with many issues such as zero-visibility and bad weather.

Automatic landing is seen as an imminent requirement in the future due to increasing air traffic requirements, such as near zero visibility landings at night and the need to automate the air cargo operations. The development of modern control and navigation systems aids the evolution of the related technologies. Classical control methods can only satisfy the requirements of a linear system. However, most real life systems are non-linear in nature; therefore, the use of the nonlinear methods in the design of an automatic landing system would be more realistically address the issue. A remarkably wide spectrum of UAV configurations is currently in use or under development, ranging from fixed and rotary wing, going from micro to jet-sized. Fixed wing UAVs are already being used in commercial applications such as rice fields spraying and fish banks search in Japan [2]. Other applications that are currently being brought forward include maintenance and security inspection of power lines, along with oil and gas pipelines, forest fire surveillance, environmental observation and monitoring of natural disaster areas, among others. Several different control theories are mainly adopted when designing controllers for autonomous landing. Fuzzy neural networks and genetic algorithm were considered by [3]. The controller was designed to command pitch angle with inputs of altitude, altitude command, altitude rate, and altitude rate command. The results from the simulations of this fuzzy neural network controller were presented as acceptable and it was mentioned that the fuzzy controllers can successfully control a vehicle in severe wind disturbances. Fuzzy logic controller (FLC) is another methodology used to control the glide path error (via altitude control), for pitch stabilization, and to maintain constant speed. The fuzzy controller uses the error (from desired path) and the time-rateof-change of the error (from desired path) to determine the control input [4]. The FLC also uses scaling factors for input variables, the error and the error rate to affect the controller response. The main benefit of this controller compared to the previous one is its simple design and its ability to account for nonlinearity. In order to design a digital auto-land controller, quantitative feedback theory (QFT) was adopted and the controller offered better overall safety, performance, and robustness for the case of an airplane with model uncertainties flying in turbulent conditions [5]. This paper utilizes the feedback linearization method to design a controller for an automatic landing system for a UAV. This method is chosen for its applicability to account for nonlinearities in the system and its simplicity. The UAV has been chosen due to its aerodynamic and propulsive data availability in published literature [6]. This control technique could be extended for application in any generic aircraft or space craft. VEHICLE DYNAMIC MODEL During landing the airplane trajectory mainly consists of two segments; the glide path and the flare. The glide path is the path the airplane adopts while descending towards the airport, the flare is an exponential path the airplane follows to adjust its attitude to a one suitable for touchdown. During both

segments the attitude of the airplane is controlled by the deflection of the elevator while the speed is controlled by the thrust. The airplane model adopted for the UAV is available in [6]. The following are the non-linear equations of motion required to simulate the basic motion of the UAV. The translational dynamics equations are derived from Newton’s second law. Both longitudinal and lateral effects are included for generality. The force equations, with the addition of gravity, aerodynamic forces, and thrust forces are given as follows
T cos( )  D cos( )  L sin( )  mg sin( )  m(v1  v32  v23 )

(1) (2)

Y  mg cos( ) sin( )  m(v2  v13  v31 )
T sin( )  D sin( )  L cos( )  mg cos( ) cos( )  m(v3  v21  v12 )

(3)

where D is the drag force, Y is the side force due to rudder, L is the lift force, and T is the thrust. The total drag equation, side-force equation, and lift equation are given by

D  CD qS Y  CY qS L  CL qS

(4) (5) (6)

where C D CY and C L are the total drag, side-force, and lift coefficients, respectively. S is the known reference area, and q is the dynamic pressure which is a function of the known air density  and velocity magnitude given as q 1 2

 v

2

(7)

The aerodynamic coefficients are given by
CD  CD  KCL
0

2

(8) (9) (10)

CY  CY  CY   CY  R  CY  A
0



R

A

CL  CL  CL   CL  E
0



E

where K,  E  R and  A are the efficiency factor, elevator (or stabilizer), rudder, and aileron angle deflections respectively. The other terms in Eq. (8, 9, and 10) are the known aerodynamic coefficients defined by the particular aircraft of interest. These reflect the contributions of the individual

quantities (e.g., CD is the drag coefficient contribution due to


angle of attack, CD is the drag coefficient for α =0).
0

The Euler equations represent the rotational dynamics, with the assumption that the thrust vector is aligned with the airplane center of gravity; the equations in component form are given by

   1 sin  tan     0 cos        0 sin  sec    

cos  tan    1 

    2 cos  sec   3   
 sin 

(21)

J 111  J 133  J 1312   J 33  J 22  23  L1

(11) (12) (13)

where the roll (  ) pitch (  ) and yaw (  ) angles can be determined by integrating the set in Eq. (21). All the aerodynamic coefficients used in this study can be found in Appendix A. LANDING TRAJECTORY The trajectory outlined by an airplane is mainly divided into two segments as motioned previously, glide path and the flare. During landing, approach and descent are initiated by intercepting the glideslope path. Zero glideslope deviation must be maintained during approach in order for the airplane to fly along the glideslope. Airspeed is maintained by power control and descent rate by angle of attack control. The glideslope error deviation is computed by using the measured glideslope angular error and the slant height of the glideslope. When the aircraft reaches a decision height, it checks for the state variables. If they are within an acceptable range, the decision is taken to continue with the landing. Otherwise, the decision is taken to delay landing by circling around. At the decision height, the control is switched over to altitude control for the flare. Glide path. The glide path is the preferred decent path of an airplane into an airport; it is outlined in Figure 1. During the glide path a beam is transmitted towards the sky at an angle  G from the ground, typical values for  G are 2.5o to 3.5o [7], this beam is received by instruments on the airplane. The difference between γ and  G is the glide path error Г; the error depends on the displacement d and the slant range R.

J 222   J11  J 33  13  J 13 1  3  L2
2 2





J 333  J 131  J 1323   J 22  J 11  1 2  L3 where L1 given by

L2 and L3 are the aerodynamic torques which are
(14) (15) (16)

L1  Cl qSb

L2  C m qSc L3  Cn qSb

where C l C m and Cn are the rolling, pitching and yawing torque coefficients respectively, b is the known wing span, and c is the known mean geometric chord. The torque coefficients are given by (17) Cl  Cl  Cl   Cl  R  Cl  A
0



R

A

Cm  Cm  Cm   Cm  E
0



E

(18) (19)

Cn  Cn  Cn   Cn  R  Cn  A
0



R

A

To determine the linear velocities with respect to the reference frame we utilize the inverse attitude matrix, which is usually defined by the 3-2-1 sequence, so that

 x   c c  y    c s     z    s   

s s c  c s s s s  c c s c

c s c  s s   v1  c c

c s s  s c   v2  (20)

    v3   

where x y and z are the velocity components with respect to the reference frame and c and s are defined as cosine and sine respectively. The aircraft’s position relative to the reference frame can be determined by integrating Eq. (20). In a similar fashion the Euler rates can be expressed using the 32-1 kinematics equations

Figure 1. Glide path schematic

Flare. The flare is the trajectory in which the flight path angle is changed to a more suitable one for the touchdown of the airplane. The height of the airplane changes in a somewhat exponential trend as shown in Figure 2.

Plugging in Eq. (18) and rearranging one obtains

 J yy   C qSc   K1  K 2    c    Cm  Cm    m  E 


(26)

Cm

E

Eq. (26) is used to reduce the glide path error to zero, where γ =

G.

The system block diagram is shown in Figure 3.

Figure 2. Flare path schematic LONGITUDINAL CONTROLLER DESIGN The method adopted for controller design is the feedback linearization method. In this method the variable to be controlled is differentiated, with respect to time, until the control variable appears in the equation. In this work, the variable being controlled during both the glide path and the flare segments is the pitch angle θ. The control variable utilized to achieve that control is the elevator deflection δE. However, during the glide path the commanded θ is acquired from the glide path error Г, which is the sum of the flight path angle γ and the glide path angle γG. During the flare segment the commanded θ is estimated through the sink rate v3 and altitude z. This is done by assuming the altitude as an exponential function of time. This section discusses in details the equations and variables required to achieve a required landing trajectory. Only longitudinal control is considered in the following study for simplicity. Glide Path Control. By referring to Figure 1, the following relationship can be obtained

Figure 3. Glide path block diagram Flare Control. The equation that governs an idealized exponential flare trajectory such as the one shown in Figure 2 is presented as follows

h  h0 e

t /

(27)

where h, h0 and τ are the height, flare entry height and time constant respectively. The distance between the point of touchdown and the flare entry height depends on the flare entry height and the velocity. Assuming the velocity is constant at 12m/s the following holds true h0  V sin   0.6 m / s (28) Diffrentiating Eq. (27) one obtains h h0 e
t /

    c
Assigning a value of 3o for pitch angle is assumed

(22)





h



(29)

G

the value of the commanded

Plugging in the value of h0 from Eq. (8) into Eq. (9) yields h0   h0  h0  0.6

 c  k  c (23)



(30)

The equation for the rate of change of θ Eq. (21) reduces to the following

  2

(24)

Eq. (24) is differentiated once more to allow for the presence of the control variable  E

The time constant τ is chosen to have a value of 5 sec. this value is relatively high, but because of the lack of attitude control during this maneuver, a more gradual flare is desirable, i.e. an exponential curve with a larger time constant [8]. Substituting this time constant into equation 10 the flare entry height is calculated to be 3m. Plugging the value of τ into Eq. (29) renders the sink rate as a function of altitude as h  0.2h

  2 

L2 J yy



Cm qSc J yy

  K1  K 2    c 

(25)

(31)

To achieve an exponential trajectory during flare the commanded pitch angle is set as follows

12m/s, and rearranging for the thrust gives the velocity controller equation as follows
T

 c   K h  h  v3   K h h
Substituting Eq. (32) into the pseudo-control in (25) gives

(32)

  K V  V   V  Da  Lb  gc c (38)

Tc

   K 3  K 4   K h  h  v3   K h h 

where (33)
Tc   a=   b

Setting the pseudo-control in Eq. (33) equal to Eq. (25) and rearranging, the equation for the elevator deflection in the flare controller is as below

 v1 cos   v3 sin    m m 
(39)

 v1 cos   v3 sin    m  m   v1 sin   v3 cos    m m 

 J yy   C qSc   K 3  K 4   K h  h  v3   K h h    Cm  Cm    E   m
Cm
E

c    v1 sin   v3 cos  

(34) The system block diagram is shown in Figure 4.

Figure 4. Flare block diagram

Velocity Control. The velocity vector is controlled by means of thrust, the equation of the velocity vector is differentiated only once until the control variable (thrust) appears, and the velocity is assumed to be constant during the landing with a value of 1.3Vstall which is about 12m/s. The velocity (V) is calculated as follows
V= v1  v 3
2 2

(35) dV dv 3 dv3 dt
 g sin   v32 

SIMULATION RESULTS A simulation is carried out using Matlab/Simulink incorporating both the dynamical model and the controllers. The landing is assumed to commence at an altitude of 20m. This altitude engages the glide path controller. When the flare entry height (3m) is reached, the flare controller is activated and ensures the exponential motion required. During the simulation, the velocity is kept constant at a value of 12m/s which is 1.3Vstall. This number is adopted for it is usually the value of an airplanes approach speed. Figure 5 shows the flight trajectory during glide path and flare. The controller’s ability to track the flight path angle during glide path and a smooth exponential trajectory during flare can be seen. It can also be seen that the flight path angle is held constant during the glide path. A smooth exponential trajectory is shown to be generated by the controller during the flare phase of the landing. Figure 6 shows the variation of the flight path angle throughout the simulation. A small oscillation is observed in the beginning but is damped to a value of -3 degrees during the glide path. When flare is initiated, the flight path angle is shown to increase in an almost exponential manner to a value of zero degrees. Figure 7 shows the controllers ability to track a velocity value of 12m/s during the simulation while Figure 8 shows the elevator deflection history during the simulation. CONCLUSION Auto-landing control systems have been widely studied in the past few decades using various methods. Many have used the linearized aircraft model and used classic controllers such as PID. Neural networks and fuzzy logic were also used in the design of auto-land systems. In this paper, a nonlinear control auto-landing system was developed using the feedback linearization technique. Simulation results show that the autolanding controller designed by the feedback linearization technique has been shown to perform well in its ability to track the flight path angle during glide path and a smooth exponential trajectory during flare. The controller performed well to keep the velocity and trajectory deviations within acceptable limits.

V=

dV dv1 dv1 dt



(36)

V=

v1  T cos   D cos   L sin 

 V

m

 

 T sin   D sin   L cos n  g cos   v     1 2  V m  v3 (37)

Setting the pseudo-control for the velocity as V= K  V-Vc  and plugging it into Eq. (36) with a commanded value of

Figure 8. Elevator deflection history Figure 5. UAV response during glide slope and flare The simulation showed that the UAV meets the criteria of safe landing. Several extensions should be included in order to simulate a more realistic environment. The effect of the actuator dynamics, aerodynamic drags of landing gear, flaps, speed-brake and the ground effect can be included. Moreover, lateral control of the UAV should also be addressed in the future in order to observe the interrelation with the longitudinal control. REFERENCES [1] Biju Prasad B., Dr. S. Pradeep, “Automatic Landing System Design using Feedback Linearization Method” AIAA Conference and Exhibit, Rohnert Park, California, 7-10 May 2007. [2] Rosa, P., Silvestre, C., Cabecinhas, D., Cunha, R., 2007. “Autolanding Controller for a Fixed Wing Unmanned Air Vehicle” AIAA Guidance, Navigation and Control Conference and Exhibit, Hilton Head, South Carolina. [3] Juang, J-G., Chin, K-C., Chio, J-Z., 2004. “Intelligent Automatic Landing System Using Fuzzy Neural Networks and Genetic Algorithm” Proceeding of the American Control Conference, pp. 5790-5795. [4] Nho, K., Agarwal, R., 2000. “Automatic Landing System Design Using Fuzzy Logic,” Journal of Guidance, Control, and Dynamics, Vol. 23, No.2, pp. 298-304. [5] Wagner, T., Valasek, J., 2007. “Digital Autoland Control Laws Using Quantitative Feedback Theory and Direct Digital Design,” Journal of Guidance, Control, and Dynamics, vol. 30, pp. 1399-1413. [6] Grankvist, Henrik. 2006. “Autopilot Design and Path Planning for a UAV” SE Scientific report. [7] Wang, R., Shen, Y., 2007. “Flying-Wing UAV Landing Control and Simulation Based on Mixed H2/H∞” Proceedings of IEEE International Conference on Mechatronics and Automation, Harbin, China. [8] Royer, D., 2010. “Design of an Automatic Landing System for the Meridian UAV using Fuzzy Logic” Master Thesis.

Figure 6. Flight path angle

Figure 7. Velocity control

[9] Crassidis, J., Junkins, J. 2004. “Optimal Estimation of Dynamic Systems” CRC Press LLC, Florida, Chap. 3 ACKNOWLEDGMENT The authors would like to thank Wahba Isam for his fruitful discussions during the preparation of this paper. APPENDIX A Table 1. UAV aerodynamic properties and other variables Notation CD0 CL0 CLα CLδ E Cm0 Cmα CmδE Ixx Iyy Izz Magnitude 0.02 0.071 3.224 0.7377 -0.0071 -0.1399 -0.5347 0.0306 0.0180 0.0251 Unit Unit-less Unit-less rad-1 rad-1 Unit-less rad-1 rad-1 kg.m2 kg.m2 kg.m2