Identification of a UAV and Design of a Hardware-in-the-Loop System for Nonlinear Control Purposes Myriam Manaï and André Desbiens LOOP, Université Laval, Quebec City, Quebec, G1K 7P4, Canada Eric Gagnon Defence R&D Canada - Valcartier, Val-Belair, Quebec, G3J 1X5, Canada a B e J f b g b i b j b k b R ned2b t b v a v ned v w w abs y M y G h m N P This article summarizes the modelling of a UAV and the identification of the model parameters in the context of a military project, aiming to develop a nonlinear control for the UAV. More precisely, the use of the output-error approach to identify the parameters that are difficult to measure is presented. Flight tests have been conducted to collect data in order to calculate the model parameters with the output-error approach algorithm. Preliminary results are satisfying since it is possible to adequately predict the roll attitude angle output for various situations. Finally, a hardware-in-the-loop system has been developed for testing purposes and is described. This system allows to connect a computer simulating the UAV model to the real UAV autopilot. Nomenclature UAV acceleration vector due to all forces except the gravitational force, expressed in body-axes Weighting matrix The quaternion of finite rotation representing the attitude of the UAV Inertia matrix Total applied force to the UAV (gravitational force excluded) in body-axes Earth s gravitational acceleration (including centripetal acceleration) in body-axes Unit vector of the X-axis of the body-axis system Unit vector of the Y-axis of the body-axis system Unit vector of the Z-axis of the body-axis system Rotation matrix expressing the orientation of the body-axis frame relative to the NED frame Total applied torque vector to the UAV Velocity of the air relative to the UAV body or UAV airspeed Groundspeed velocity Wind velocity, assumed to be constant UAV absolute angular velocity vector in body-axes Predicted model output vector Recorded UAV output vector Least-squares sum criteria Altitude above the take off point UAV mass Number of recorded samples UAV roll angular velocity Master degree student, Department of Electrical and Computer Engineering, Pavillon Adrien-Pouliot, AIAA nonmember Professor, Department of Electrical and Computer Engineering, Pavillon Adrien-Pouliot, AIAA nonmember Defense Scientist, Precision Weapon Section, 2459 Pie-XI Blvd North, AIAA nonmember 1 of 6
Q R Subscripts i Symbols λ l φ φ l ψ θ UAV pitch angular velocity UAV yaw angular velocity i th sample Geodetic latitude Roll attitude angle or bank angle Geodetic longitude Yaw attitude angle or heading Pitch attitude angle or elevation angle I. Introduction The purpose of this project is to modify the autopilot of an existing aerial platform to achieve the performances required by military applications. The existing autopilot is based on gain-scheduled PID controllers. The main objective is to replace some PID controllers by a nonlinear control algorithm based on a physical model of the UAV. The first step consists in building a UAV model and to identify its parameters. The UAV model is nonlinear with six-degrees-of-freedom and is simulated with Simulink. This model is based on an existing library and has been slightly modified for our purposes. For more details about this library, refer to the Aerosym library user s guide. 1 The model expresses aerodynamic forces and moments as a sum of several terms using stability and control derivatives. The identification of the stability and control derivatives of the model and of some other model parameters difficult to measure is accomplished using the output-error approach. Finally, for testing purposes, a system allowing to first control the computer UAV model with the purchased autopilot or with a nonlinear algorithm implemented in the autopilot has been developed. This system is refered to as the hardware-in-the-loop system and will be useful in the next steps of the project. II. The UAV and its on-board Autopilot An MP2028 autopilot and a Xtra Easy2 UAV were both purchased from Micropilot. The Xtra Easy2 is a fixed-wing UAV developed by Hangar 9, with a wingspan of 1.75m, a wing area of 0.5m 2 and an overall weight varying between 2.8Kg to 3.2Kg. The MP2028 is equipped with several sensors, enumerated in table 1. This autopilot was on-board the UAV during the flight tests and its sensors were used to collect data required for the identification of the UAV. Furthermore, in futur works, a nonlinear control will be implemented within this autopilot. An MP2000 autopilot without any sensors was also purchased and used to develop the hardware-in-the-loop system. The MP2000 autopilot is the previous version of the MP2028 autopilot. III. Six-Degrees-of-Freedom UAV Dynamic Model The UAV is represented by a nonlinear six-degrees-of-freedom model available from the Aerosym library, from Unmanned Dynamics. This model uses both of the following common coordinate systems. The body-axis system is a right-handed orthogonal axis system fixed to the UAV. The center of this axis system is located at the center of gravity of the UAV. The axes follow some convenient reference lines of the UAV body. As seen by the pilot, the X-axis follows the fuselage and points foward, the Y-axis points through the right wing and the Z-axis points downwards. Unit vectors of the X, Y and Z axes are respectively i b, j b and k b. The north-east-down axis system (NED frame) is a right-handed orthogonal axis system located on the surface of the Earth, under the UAV. The X-axis is directed toward the north, the Y-axis points toward the east and the Z-axis points downwards. The XY plane is always tangent to the Earth s surface and the entire axis system moves with the UAV. 2 of 6
In the model, the force and moment equations are expressed in the body-axis system. In this axis system, the UAV absolute angular velocity vector is: w abs = P i b + Qj b + Rk b (1) where P is the roll angular velocity, Q is the pitch angular velocity and R is the yaw angular velocity. It is possible to consider the NED frame on the surface of the Earth as an inertial reference frame, leading to the flat-earth state equations. Altough this frame is rotating, the angular velocity of the Earth through the force equation development is neglected, except for the centripetal acceleration due to the Earth rotation. By doing so, the force equation is developed with the UAV relative velocity vector with respect to the Earth and leads to the following expression for the time variation of the relative velocity vector in body-axes, considering the UAV mass is constant: 2 v b = 1 m f b w abs v b + g b (2) where f b is the total applied force vector to the UAV except the gravitational force and is a sum of the aerodynamic force vectors and the propulsion force vector, m is the mass of the UAV and g b is the Earth s gravitational acceleration vector (including centripetal acceleration), all expressed in body-axes. Still in the body-axis system, the moment equation is developed with the absolute angular velocity vector of the aircraft and leads to: 2 ẇ abs = J 1 (w abs (Jw abs )) + J 1 t b (3) where t b is the total applied torque vector to the UAV and J is the inertia matrix. Since the UAV is symmetrical with respect to the XZ plane of the body-axis system, only the Jxz cross-product has a nonzero value. The total applied torque vector is a sum of the aerodynamic moment vectors, the propulsion moment vector and the moment vectors due to the fact that aerodynamic forces are not applied directly to the center of gravity, but rather to the UAV aerodynamic center. 1 Determining the attitude of the UAV consists in computing the orientation of the body-axis reference frame relative to the inertial reference frame (NED frame). The rate of change of the attitude of the UAV is computed from its angular velocities P, Q and R. To do so, the Euler-Rodrigues quaternion formulation is used, because it allows avoiding some singularities present in the equations using the Euler angles. The rate of change of the attitude is thus represented by the rate of change of the components of the quaternion e, given by the following equation: 2, 3 ė 0 ė x ė y ė z = 1 2 0 P Q R P 0 R Q Q R 0 P R Q P 0 The rate of change of the quaternion components can be integrated to obtain the quaternion. Then, the quaternion components can be used to get the Euler angle rates of change: 3 φ θ ψ = e 0 e x e y e z atan2[2(e 0 e x + e y e z ), (e 0 2 + e 2 z e x 2 e y 2 )] asin[2(e 0 e y e x e z )] atan2[2(e 0 e z + e y e x ), (e 0 2 + e 2 x e y 2 e z 2 )] (4) (5) where φ is the roll attitude angle, θ is the pitch attitude angle and ψ is the yaw attitude angle. By integrating the Euler angle rates of change, the Euler angles are obtained. The UAV groundspeed vector expressed in the NED frame v ned is obtained from a rotation matrix expressing the attitude of the body-axis frame relative to the NED frame. The rotation matrix R ned2b is built from the Euler angles, but the Euler-Rodrigues quaternion formulation could be used equally: 3 v ned = R ned2b T v b (6) The geodetic position in term of latitude, longitude and altitude are obtained from the groundspeed vector components and from some Earth s parameters. 4 The Earth s parameters used through the model 3 of 6
equations, such as gravity and radii of curvature along lines of constant longitude and latitude, come from the World Geodetic System WGS-84. 4 Equation (2), Eq. (3), Eq. (4), and Eq. (6) are the state equations defining the flat-earth model. The aerodynamic forces considerated in the total applied force vector f b used in Eq. (2) are lift, drag, and side forces. The aerodynamic moments in the total applied torque vector t b used in Eq. (3) are pitch, roll, and yaw moments. In general, an aerodynamic force or moment can be computed from a nondimensional coefficient. According to the small-disturbance theory, each of the nondimensional force and moment coefficients are represented by small-perturbation equations and are a sum of several terms formed by nondimensional 2, 5, 6 stability and control derivatives. The aerodynamic force and moment coefficients are function of the attack angle, the sideslip angle, the actuator angles (elevator, rudder, and ailerons), and the airspeed. All force and moment equations used through the model are detailed in the Aerosym user s guide. 1 The model inputs are the actuator angles and the throttle fraction needed to compute the aerodynamic and propulsion forces and moments. The model provides outputs corresponding to all measures generated by the UAV sensors, as detailed in table 1. The acceleration vector of the aircraft due to aerodynamic and propulsion forces is obtained as follows: 1 and the airspeed vector v a is given by: 1 where v w represents the wind vector that is assumed to be constant. a = f b m = a xi b + a y j b + a z i b (7) v a = v b v w (8) Table 1. MP2000 and MP2028 sensors and the corresponding outputs of the model. Sensors Signals Model outputs Two accelerometers * X and Y body-axes accelerations a x and a y A differential pressure sensor Airspeed in the X-axis v ax An altimeter Altitude above the take off point h Three rate gyros Roll, pitch and yaw angular rate P, Q and R A GPS Geodetic position and groundspeed λ l, φ l, h and v ned * The MP2028 has a third accelerometer providing the Z body-axis acceleration (the Z body-axis acceleration is neither used in the identification nor in the hardware-in-the-loop system). IV. Identification of the UAV Model Parameters The UAV model has several parameters for which values need to be estimated. Parameter values of the model have to be chosen carefully to adequately represent the behavior of the UAV. Easily measurable parameters are determined by measures. This is the case for the mass of the UAV, its inertia moments and some of its physical dimensions such the wingspan, the mean aerodynamic chord, and the wing area. Other parameters are determined with the output-error approach algorithm, using data obtained from flight tests. 7 9 The parameters estimated with this method are the stability and control derivatives from which each aerodynamic coefficient is evaluated, that is drag force, lift force, side force, pitch moment, roll moment and yaw moment coefficients. Also, the output-error approach is used to find the x position of the aerodynamic center and the J xz inertia product. A. The Output-Error Approach The process to be identified is excited by manipulating the inputs while the inputs and the outputs are recorded. Then, the model parameters are estimated by minimizing the follwing cost function: G = N [y M (i) y(i)] T B [y M (i) y(i)] (9) i=1 4 of 6
where N is the number of recorded samples, B is a weighting matrix, y(i) is the recorded process output vector and y M (i) is the predicted model output vector for the given process inputs. For better results, the inputs should be rich at the frequencies where precision is necessary. Various input amplitudes are also required in presence of nonlinearities. The resulting model should be validated with data that were not used for identification. B. Experiment An experienced pilot has executed some flights with the UAV. During each flight, command and sensor data were sampled and recorded in the autopilot memory. At the end of the flight, data samples were downloaded in a computer and saved in a data file for identification. The inputs are the commands given to the UAV, i.e. the throttle fraction, the elevator angle, the rudder angle, and the ailerons angle. The outputs are all of the sensors listed in table 1, the altimeter excluded. The output-error approach algorithm has been implemented with Matlab, using the UAV Simulink model. Constraints are taken into account during the mininization of the criteria G. Maximum and minimum values for each parameter are imposed to avoid unstability. Furthermore, some constraint equations were implemented to avoid some parameter combinations for which the model becomes unstable. It is important to provide appropriate initial parameter estimates in order to be as near as possible of a solution. For the first attempt, initial estimates are chosen following a reference book 5 and from the Aerosonde UAV example provided with the Aerosym library. 1 A next step, as an improvement, will be to provide the initial estimates from methods contained in the Datcom handbook. 10 Roll attitude angle, degrees 80 60 40 20 0 20 40 60 80 Roll data from the model Roll data from the flight test 100 0 20 40 60 80 100 120 Time, s Figure 1. The roll attitude angle output of the model with the parameters identified by the error-output approach algorithm, along with the roll attitude angle measured during the flight test. C. Results The roll attitude angle output of the model with the parameters identified by the output-error approach algorithm is shown in figure 1, along with the roll attitude angle measured during the flight test. The fit is good enough for an eventual nonlinear controller design. V. Hardware-in-the-Loop System The hardware-in-the-loop system consists in connecting the autopilot to a computer running the UAV model. The computer plays the role of the UAV and the autopilot can control the model, instead of the real UAV. The UAV model, implemented with Simulink and based on the identification results, is compiled with Real Time Workshop of Simulink and is downloaded in the computer in order to allow the computer to run the model in real time. The hardware-in-the-loop system is illustrated in figure 3(a). In the hardware-in-the-loop system, the simulator reads the command signals sent by the autopilot to the actuators, namely the elevator, the ailerons, the rudder, and the throttle. Then, the simulator traduces them in actuator angles and in throttle fraction, and provides them to the UAV model. Finally, the simulator generates the sensor signals by executing the UAV model with the actuator angles and throttle fraction provided as input. Therefore, the simulator includes the UAV model plus other Simulink blocks that we created in order to reproduce the sensor outputs and to interpret the autopilot commands. The computer is connected to the autopilot through digital and analog acquisition cards. An interface has been conceived to allow a proper communication between the autopilot and the UAV model. This interface is represented in details in figure 3(b). A first part of the interface implementation is software, as a part of the simulator. A second part of the interface is hardware and includes the acquisition cards and some printed circuit boards. The hardware-in-the-loop system can be detailed as follows. The autopilot commands, sent as puslewidth modulation () signals, are read through an NI6601 digital acquisition card. To interpret the 5 of 6
Figure 2. The hardware-in-the-loop system. MP2000 Servo commands: Throttle Ailerons Rudder Elevator UAV simulator Command translation UAV model Hardware PCBs PCI card PCI D/A card Data conversion Voltage Software Model I/O Accelerations Airspeed Altitude Angular velocities RS232 GPS paquets GPS positions GPS groundspeed RS232 Sensor simulation Throttle Ailerons Rudder Elevator PCI card Actuator angles Throttle position (a) General system. (b) Detailed interface. command signals, the pulse widths are then converted in actuator angles and throttle fraction in Simulink. To reproduce the sensor outputs, the model outputs are converted in Simulink under the same form as the sensor outputs, that is signals, analog voltages and GPS paquets. Then, the signals are sent to the autopilot through an NI6601 digital aquisition card and voltage signals are transmitted through an NI6703 analog acquisition card, while the GPS paquets are sent to the autopilot via RS232 protocol. For some voltage signals, the resolution of the NI6703 card is not high enough to represent properly the sensor output for all its range of values. For these signals, only one part of the model outputs is converted in software to reproduce the sensor outputs, the other part is accomplished with printed circuit boards that we conceived specially for this task. VI. Conclusion In this article, we present the results obtained for the identification of the UAV model parameters with the output-error approach for one model output, the roll attitude angle curve. In near futur works, we aim to identify the UAV model parameters with this method so that all model outputs enumerated in table 1 be comparable to the UAV sensor outputs recorded during flight tests, in order to obtain a representative model of the entire UAV behavior. Also, we describe a hardware-in-the-loop system. This system will be useful for testing new control algorithms with the UAV model before attempting to apply the control algorithms to the real UAV. It will also be useful for evaluating the performance of the existing autopilot and for analysing the UAV model behavior. References 1 Unmanned Dynamics LLC, Hood River, AeroSym Aeronautical Simulation Blockset, Version 1.1, User s Guide, 2003. 2 Stevens, B. L. and Lewis, F. L., Aircraft Control and Simulation, John Wiley & Sons, Inc., 1st ed., February 1992. 3 Phillips, W. F., Hailey, C. E., and Gebert, G. A., Review of Attitude Representations Used for Aircraft Kinematics, Journal of Aircraft, Vol. 38, No. 4, August 2001, pp. 718 737. 4 Rogers, R. M., Applied Mathematics in Integrated Navigation Systems,, Inc., Virginia, July 2000. 5 Blakelock, J. H., Automatic Control of Aircraft and Missiles, John Wiley & Sons, Inc., USA, 2nd ed., January 1991. 6 Etkin, B. and Reid, L. D., Dynamics of Flight, Stability and Control, John Wiley & Sons, Inc., 3rd ed., January 1996. 7 Maine, R. E. and Iliff, K. W., Application of Parameter Estimation to Aircraft Stability and Control, the output error approach, NASA Reference Publication 1168, National Aeronautics and Space Administration, Scientific and Technical Information Branch, June 1986. 8 Ljung, L., System Identification - Theory for the User, Prentice Hall, Upper Saddle River, N.J., 2nd ed., January 1999. 9 Jategaonkar, R. V. and Thielecke, F., Aircraft Parameter Estimation - A Tool for Development of Aerodynamic Databases, Sādhanā, Vol. 3, No. 2, April 2000, pp. 119 135. 10 Hoak, D. E. and Finck, R. F., USAF Stability and Control Datcom, Flight Control Division, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, April 1978. 6 of 6