CHAPTER 6 DESIGN OF SIX DEGREES OF FREEDOM AIRCRAFT MODEL AND LONGITUDINAL AUTOPILOT FOR AUTONOMOUS LANDING

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1 148 CHAPTER 6 DESIGN OF SIX DEGREES OF FREEDOM AIRCRAFT MODEL AND LONGITUDINAL AUTOPILOT FOR AUTONOMOUS LANDING 6.1 INTRODUCTION This chapter deals with the development of six degrees of freedom (6-DOF) aircraft model. This 6-DOF model can be used to design the longitudinal autopilot for autonomous landing. Glide slope and flare autopilots are designed and implemented using the 6-DOF model. Finally the results are verified using X-Plane Flight simulator. 6.2 DEVELOPMENT OF 6-DOF AIRCRAFT MODEL The block implements a nonlinear 6-degree-of-freedom aircraft dynamic model, using blocks provided in the AeroSim library (Aerosim Aeronautical Blockset - User s Guide). The equations of motion are implemented in geodetic-frame. The model parameters are read from a userconfigurable mat-file. The 6 DOF aircraft model block is shown in Figure 6.1. The various parameters given as inputs and the outputs obtained are discussed in the Section The complete aircraft model is shown in Figure 6.2.

2 149 Figure DOF Aircraft Model Block Block characteristics The Parameters of 6-DOF Aircraft Model Block are given below: Aircraft configuration file: The path and name of the aircraft parameter matfile, provided as a string. For example, if the mat-file is someairplane.mat, and it is in the current directory, then we would use someairplane.mat. Initial position: The 3 1 vector of initial aircraft location [Lat Lon Alt ]T, in [rad rad m]. Initial velocities: The 3 1 vector of initial aircraft velocity components in geodetic-frame [V N V E V D ]. Initial attitude: The 4 1 vector of initial aircraft attitude provided as Euler- Rodrigues quaternion s [e 0 e x e y e z ]. Initial angular rates: The 3 1 vector of initial aircraft angular rates (in body axes) [ p q r ]. Initial fuel mass: The initial mass of the fuel quantity available on-board the aircraft, in kg.

3 150 Initial engine speed :The initial engine shaft rotation speed, in rad/s. Ground altitude: The altitude of the terrain relative to mean-sea-level, at aircraft location, in meters. WMM coefficient file: The complete path to the magnetic model coefficient file, Simulation date: 3 1 vector of the calendar date in the format [Day Mon Year]. Sample time: The sample time at which the aircraft model will run. The inputs of 6-DOF Aircraft Model Block are given below: Controls: The 7 1 vector of aircraft controls [flap elevator aileron rudder throttle mixture ignition ] in [rad rad rad rad frac ratio bool]. Winds: The 3 1 vector of background wind velocities, in navigation frame [WN WE WD], in m/s. RST: The integrator reset flag The outputs of 6-DOF Aircraft Model Block are given below: States: The 15 1 vector of aircraft states [V N V E V D p q r e 0 e x e y e z Lat Lon Alt mfuel Weng ]. Sensors: The 18 1 vector of sensor data [Lat Lon Alt V N V E V D a x a y a z p q r p stat, p dyn OAT H x H y H z ]. VelW: The 3 1 vector of aircraft velocity in wind axes [Va b a] in [m/s rad rad]. Mach: The current aircraft Mach number.

4 151 Angular Acc: The 3 1 vector of body angular accelerations [p q r ]. Euler: The 3 1 vector of the attitude of the aircraft given in Euler angles [f q y], in radians. AeroCoeff : The 6 1 vector of aerodynamic coefficients [C D C Y C L C l C m C n ], in rad. PropCoeff : The 3 1 vector of propeller coefficients [J C T C P ]. EngCoeff: The 5 1 vector of engine coefficients [MAP m air m fuel BSFC P]T given in [kpa kg/s kg/s g/(w*hr) W]. Mass: The current aircraft mass, in kg. ECEF: The 3 1 vector of aircraft position in the Earth-centered, Earth-fixed frame [X Y Z ] MSL: The aircraft altitude above mean-sea-level, in m. AGL: The aircraft altitude above ground, in m. REarth: The Earth equivalent radius, at current aircraft location, in m Complete Aircraft Model Sub Block The complete aircraft sub block consists of the aerodynamics block, propulsion system, aircraft inertia, the atmosphere model, total acceleration, the total moment sub blocks, the aircraft equations of motion and the earth model. The earth model implemented here is the WGS-84 model. The acceleration, velocity, the rates and the position are calculated using the Equations of Motion sub block. The initial conditions i.e. the flap, elevator, rudder, throttle, ignition, mixture are provided to the aerodynamics and the propulsion block. The initial wind condition is given to the

5 152 atmosphere block. The initial velocities, rates and lat, lon, alt position are given in the Equation of Motion (EOM) block. Hence the complete aircraft model block is shown in Figure 6.2. Figure 6.2 Complete Aircraft Model Sub Block

6 Aerodynamics Block Aerodynamics: Section 6.21 of the aircraft con guration script speci es the aerodynamic parameters of the aircraft. These are explained in the following Figures Figure 6.3 Aerodynamics Sub Block

7 154 Aerodynamic parameter bounds the limits that the aircraft model will impose on the airspeed, sideslip, and angle-of- attack, given as 1x2 vector of min and max values. The purpose of using these limits is to keep the outputs of the aerodynamic model within the linear region Propulsion Block Figure 6.4 Propulsion Sub Block Propeller : The second section of the aircraft con guration script speci es the geometry and aerodynamic performance of the propeller. Propeller hub location: The position of the propulsion force and moment application point, given with respect to the body- frame origin. The location is speci ed as a 1x3 row vector of x, y, and z coordinates. Advance ratio: The aerodynamic performance of the propeller should be given as a look-up table of propeller coef cients (CP and CT) as functions of

8 155 the propeller advance ratio. This variable speci es the advance ratio vector which corresponds to the look-up table. Coef cient of thrust: The vector of coef cients of thrust for the advance ratios given above (the vector should have the same size). Coef cient of power: The vector of coef cients of power for the advance ratios given above (the vector should have the same size). Propeller radius: The radius of the propeller is used by the propulsion model to compute the force and torque from the normalized coef cients. Engine : The third section of the aircraft con guration scripts allows the user to specify the engine characteristics. All engine data is given at sea-level. The engine model will correct the data for altitude effects. For a normallyaspirated general aviation piston engine, this includes the following parameters: RPM: The vector of engine speeds for which the engine data is given, in rotations-per-minute. All engine parameters are speci ed as 2-D look-up tables (functions of engine speed and intake manifold pressure). Fuel ow: The sea-level fuel ow as a function of RPM and MAP. The number of rows in the matrix should match the size of the RPM vector, the number of columns should match the size of the MAP vector. Power: The engine power at sea-level, as a function of RPM and MAP. The number of rows in the matrix should match the size of the RPM vector, the number of columns should match the size of the MAP vector. Sea-level atmospheric conditions: The sea-level atmospheric conditions, including pressure in Pascals and temperature in degrees Kelvin, for which the engine data above is given.

9 156 Engine shaft inertia: The moment of inertia of the rotating parts of the engine. This is added to the propeller inertia and used in the propulsion equation of motion to compute the current engine speed. Generally, the engine shaft inertia is signi cantly lower than that of the propeller, and it can be neglected without any major effects over the aircraft dynamics Atmosphere Block Figure 6.5 Atmosphere Sub Block The standard atmosphere block provides the air parameters at the current altitude. The standard atmosphere block is using interpolation through look-up tables which provide air data for an altitude range of 0 to meters. The background wind block computes the background wind velocity components in body axes. The block is applying a frame

10 157 transformation from inertial (geographic) to body frame, using the rotation matrix provided. The numerical time derivative of the resulting velocity vector is then computed. The turbulence block provides a von Karman turbulence model. The block is applying von Karman turbulence shaping filters for longitudinal, lateral, and vertical components to 3 white-noise sources. The filter parameters depend on background wind magnitude and current aircraft altitude. The wind shear block computes the angular rate effects caused by the variation in time/space of the background wind and turbulence velocities. The wind shear effects considered are the angular velocities and accelerations for pitch and yaw Earth Model The Earth library folder includes blocks that model the Earth s shape, gravity, and magnetic eld as shown in Figure 6.6. WGS-84: The block computes the local Earth radius and gravity at current aircraft location using the WGS-84 Earth model coefficients. EGM-96: The block computes the sea-level altitude with respect to thewgs- 84 ellipsoid, using the EGM-96 geoid undulation model. The EGM-96 block computes the altitude difference between the theoretical ellipsoid shape and the actual mean sea level (geoid undulation). This is caused by the nonuniformity of Earth s gravitational potential. The correction is performed using a 2-dimensional Latitude-Longitude look-up table with a resolution of 1 degree in both directions. The geoid undulation is then added to a 0.53 m WGS-84 correction and to the WGS-84 altitude computed by the aircraft equations of motion, to obtain the altitude of the aircraft above sea-level.

11 158 Figure 6.6 Earth Model Sub Block Ground Detection: The Ground Detection block computes the aircraft altitude Above Ground Level and sets a flag if it is zero. The ground altitude should be supplied by the user as a constant or a look-up table of terrain elevation data. In both cases it should be measured with respect to the MSL and the unit of measure must match that of the MSL altitude. WMM-2000: The WMM-2000 block computes the Earth magnetic eld components at current location using the Department of Defense World Magnetic Model Body Frame EOM These equations form the centerpiece of an aircraft dynamic model and implemented in the MATLAB as shown in Figure (6.7). There are two

12 159 formulations for the equations of motion that are commonly used, and they are provided in two separate sub-folders within the AeroSim library. Figure 6.7 Body Frame EOM Block DOF AIRCRAFT SIMULATION Using the AeroSim blockset a basic 6-DoF model is constructed as shown in Figure 6.8. In the first case a simple closed loop configuration is analysed and the airspeed and pitch angle output are obtained as shown in Figure 6.9. Using this six DoF model the landing autpilot is being implemented.

13 Figure 6.8 AeroSim 6-DOF Simulink Model without PID 160

14 161 Figure 6.9 Airspeed and Pitch angle output without PID In the second case, as shown in Figure 6.10, a PID control is added to the feedback loop which atabilised the airspeed and the pitch angle outputs.

15 Figure 6.10 AeroSim 6-DOF Simulink Model with PID 162

16 163 Figure 6.11 Airspeed and Pitch angle output with PID As seen from the Figure 6.10 and Figure 6.11 the response of the airspeed and pitch angle has improved with the addition of the PID controller in terms of settling time, overshoot and rise time. Using this 6 DoF mathematical model developed using Aerosim blockset in MATLAB/Simulink, an autonomous landing controller is designed. The requirements to successfully complete an autonomous landing are: define the glide path and flare path geometry, design the pitch autopilot and design controllers for glide path and flare. 6.4 DESIGN OF PITCH AUTO PILOT To begin the design of the pitch angle autopilot, a transfer function representative of the UAV in landing conditions is required. The aircraft

17 164 transfer function for pitch angle for the experimental UAV has been found to be: e 1728s s s s Figure 6.12 Pitch autopilot without PID Controller Figure 6.13 Pitch autopilot and its response The response, for a unit step input, shows that the oscillations are more and is not quite good for landing, since the settling time and rise time are more. The actual settling time of 17 seconds is much too slow for an autopilot to control an aircraft on landing. In order to decrease the rise time, a

18 165 proportional controller is needed. In order to reduce the settling time, the derivative controller is needed. In order to reduce oscillations, the integrator is needed. So, overall, there is a need for PID controllers as shown in Figure The tuning of Kp, Kd, Ki are done by Hessian Modified tuning to get the optimum response. Figure 6.14 Pitch autopilot with PID Controller Figure 6.15 Pitch angle autopilot with PID controllers From the Figure 6.15, it is concluded that the proportional, integral and derivative controller gains are tuned to get the optimum response and the values are found to be Kp= 2.5, Ki= 5, Kd= That is, the system will not undergo unstable region with change in gain. The phase margin is found to be 33.8 degree and a crossover frequency of 1.57 rad/s with a settling time around 4 seconds.

19 AUTOMATIC GLIDE SLOPE CONTROLLER Basic Longitudinal Autopilot The Basic longitudinal Autopilot (John Blakelock 1991) ensures that the pilot pitch angle command is achieved quickly with fewer oscillations. It is a modified simple displacement autopilot and shown in Figure ref S (amp) e a e e Elevator Servo A/ C Rate Gyro Figure 6.16 Basic Longitudinal Autopilot It was modified using an inner loop with pitch rate feedback to improve the damping of the short period oscillations and also to achieve higher damping in outer loop Glide Slope Controller The Automatic glide slope controller guides the UAV down a predetermined glide slope of 5.5. At a pre-selected altitude reduces the rate of descent and cause the UAV to flare out and touch down with an acceptably low rate of descent (Kim and Golnaraghi 2004). The glide path is defined as a line from some starting point to the end of the runway. For this project, a glide path angle of -5.5 o was used, so the starting point was defined by the LLA position of the end of the runway and the desired final approach distance. To simulate this dependence on the range from the aircraft to the runway, the glide path command signal was defined to

20 167 include the range. Figure 6.17 shows the glide path geometry where the commanded height above ground is a function of the range. R H Runway x Figure 6.17 Geometry of the glide slope During the flare maneuver, pilots transition from flying a straight line to an exponential path to slow the descent rate of the airplane. This can be simulated by defining an exponentially decaying flight path and using altitude above ground to generate the error signal to the controller. Figure 6.18 shows the flare path geometry intended touchdown zone approximately 500 ft. from the runway threshold. H Runway x Figure 6.18 Geometry of the flare path If the UAV is below the center line of the glide slope, then d is considered negative, as is,, when the velocity vector is below the horizon, that is, the UAV descending.

21 168 The component of forward velocity U perpendicular to the glide slope center line is d and this d for small glide slope angle is given by U (6.7) 57.3 o o d Usin( 5.5) ( 5.5) If <5.5 o, then from Figure 6.17, ( +5.5) o is positive; therefore d is positive, and as d initially was negative, the UAV is approaching the glide path from below. U d ( 5.5) 57.3s o (6.8) The glide slope receiver does not measure the perpendicular distance to the glide slope centerline but the angular error resulting there from. Thus for a given value of d the angular error increases as the UAV nears the runway, which has the effect of increasing the system gain as the range to the runway decreases. For small angles, o (57.3d / R) (6.9) Through the use of Equations (6.8) and (6.9), the flight path angle can be related to the angular error of the UAV from the centerline of the glide slope. The block diagram of the glide slope control system, including the geometry is shown below in Figure ref = 0 comm o d Coupler UAV & d Autopilot 57.3 R 3 o Figure 6.19 Block diagram of glide slope control system

22 DESIGN OF GLIDE PATH CONTROLLER The glide slope command is designed in such a way that the controller will generate the error signal by comparing the instantaneous height of the UAV, obtained from the image, and the required height at that instant. So the error signal drives the autopilot, thus making UAV to align with the glide path. Figure 6.20 shows the glide path controller. Figure 6.20 Glide Path controller The Range and the instantaneous height are extracted from the runway image taken at real time. Since the sine of small value is approximately a small value, so sin(theta) = Theta. This glide path command generates the necessary error signal if the UAV misses the actual glide path. 6.7 DESIGN OF FLARE PATH CONTROLLER As seen earlier, the flare path geometry is an exponential one. After certain distances, its need for switch over from glide path to flare. The flare path command is proportional to the difference between horizontal distance at

23 170 the instant of start of flare phase and instantaneous horizontal distance from runway threshold. Let it be x. From Equation (6.10), 0 x H He (6.10) The desired value of Tau can be obtained by specifying the distance to the touchdown point from the glide slope transmitter. If this distance is to be 10 ft and if it is also assumed that the aircraft touches down in single time constant, then the ground distance traveled during the flare will be greater than 10 ft. Let the starting flare path geometry range would be 20 feet. Hence the Tau value been estimated as 1 and the implementation is shown in Figure Figure 6.21 Flare path controller Using MATLAB/Simulink basic longitudinal autopilot shown in Figure 6.22 and a Glide Slope Control system shown in Figure 6.23 are designed to guide the UAV along a predefined path having a slope of 5.5 o and maintaining the longitudinal attitude.

24 171 Scope5 Scope4 180/pi Constant1 Step PID PID Controller -10 s+12 elevator Servo -1.39s s s Thetadot / delta e 1 s Integrator theta_in thetadot_in theta phi psi thetadot Product Scope vision data calc thetafinal To Workspace5 Scope3 Figure 6.22 Basic Longitudinal Autopilot Figure 6.23 Glide Slope Control System The vision data calculation module calculates the pitch angle components and it uses the current pitch angle given by the (s)/ e (s) transfer function to set the current orientation of UAV as shown in Figure Using this orientation and current position, images are generated with a separation of 0.5 sec (Rives and Azinheira 2002). The final pitch angle estimated is fedback for the closed loop. The basic longitudinal autopilot is used in the Glide Slope control system, which guides the UAV along a predefined glide path.

25 Figure 6.24 Vision Data Calculation Module 172

26 173 Figure 6.25 Pitch angle It is concluded from Figure 6.25 that the UAV is aligned to the runway and follows the predefined glide angle (5.5 o ) to safely land the UAV. 6.8 BLENDING FUNCTION AND ITS IMPLEMENTATION An additional problem in the flare controller design was the method of switching from glide slope to flare command signals. With the switch at a range, the aircraft could not be made stable as having a switch in the simulation caused adverse affects on the input signals to the switch. It was determined by experiment that constants in Simulink create problems in solving the algebraic loop and cause the simulation to produce erroneous results. To compensate, step blocks were used with the step value equal to that of the desired constant and the step time equal to the first time step of simulation. Even with this correction, stability problems were still evident at the switch of control commands. To compensate for the sudden switch in commands, a blending function was developed to soften the effect of the switch. This function blends the signals over range values of 50 m to 20 m. These values were selected to ensure that the aircraft would be established under the flare controller before reaching the desired switch range of 50 m. Figure 6.26 shows the Simulink of the braking function. The saturation blocks normalize the signal multipliers to a value between zero and

27 174 one. 20 m is subtracted from the range and multiplied by a gain to generate a multiplier for the glide path command. The gain was selected such that at 50 m range, the glide multiplier is one and at 20 m range, the glide multiplier is zero. The flare multiplier is much the same, but in the reverse direction. These multipliers directly multiply the glide and flare signals such that when the range is between 50 m and 20 m, both signals are active. Above 50 m only the glide path signal is active and below 20 m only the flare signal is active. Figure 6.26 Glide/Flare Blending Function The blending of signals at the switch of glide slope and flare control signals solved the problem of extreme oscillation and instability during the switch. Because the flare path command is exponential, the aircraft tends to bounce if the elevator remains under control of the flare controller after main gear touchdown. To ensure the aircraft remains on the ground, the elevator is neutralized with a relay at an altitude above ground level equal to the height of the main gear. 6.9 INTEGRATION OF GLIDE/FLARE CONTROLLERS WITH PITCH AUTOPILOT Figure 6.27 shows the complete autonomous landing autopilot. Here, the autopilot engages with glide controller till the altitude of 50 m. Between the altitude ranges of 50 m and 20 m, both glide and flare controllers are effective. Below the altitude range of 20 m, the autopilot engages with flare controller. The blending function is to smooth out the switching between glide and flare phase change.

28 Figure 6.27 Glide/Flare autopilot 175

29 BRAKING FUNCTION Upon main gear touchdown, the elevator must allow the nose of the airplane to rotate downward to contact the nose gear with the ground and the brakes must be applied. Because exponentially decaying functions never actually reach zero, the elevator command must be switched from the flare path to neutral upon main gear touchdown.the brakes must be applied smoothly after touchdown or the gear will fail. A rate limiter after a switch can be used to accomplish this. Because the flare path command is exponential, the aircraft tends to bounce if the elevator remains under control of the flare controller after main gear touchdown. To ensure the aircraft remains on the ground, the elevator is neutralized with a relay at an altitude above ground level equal to the height of the main gear. At the same altitude, the throttle command is also neutralized and the brake command is changed from one to zero (zero to full braking) with a rate limiter to limit the brake application time to two seconds. The rate limiter on braking prevents gear failure due to over-braking upon touchdown. Figure 6.28 Braking Command 6.11 SIMULATION RESULTS To check the performance of the INS/Vision algorithm it was interfaced with the flight simulator X-Plane and the snapshots at different stages of the flight are shown in Figures 6.29 to 6.34.

30 177 Figure 6.29 X-plane output - Airplane to align with the runway Figure 6.30 X-plane output - Airplane at the starting of the glide slope

31 178 Figure 6.31 Airplane after glide slope and flare X plane output - Front view Figure 6.32 Airplane landed properly after glide slope and flare X plane output - Front view

32 179 Figure 6.33 X-plane output showing the actual and the estimated path of the aircraft Figure 6.34 X-plane output showing the path of the aircraft

33 180 Figures 6.29 to 6.34 clearly demonstrates that the UAV is successfully and safely landed with the integration algorithm output CONCLUSION In this chapter, a non-linear 6-DOF model is developed using the Aerosim blockset in MATLAB software and the longitudinal autopilot with glide and flare controllers are designed to control the landing of the UAV. The autonomous landing is demonstrated with the integrated vision and SDINS data and validated by interfacing the autopilot output with the X-plane simulation software. The generated control signals are transferred to X-Plane through UDP from the Simulink and the landing performance was verified and found to be acceptable.