An Adaptive Flight Control system for a Flapping wing ...

AIAA SciTech Forum 8–12 January 2018, Kissimmee, Florida 2018 AIAA Guidance, Navigation, and Control Conference

10.2514/6.2018-1836

An Adaptive Flight Control system for a Flapping wing Aircraft

Downloaded by WICHITA STATE UNIVERSITY on February 7, 2018 | http://arc.aiaa.org | DOI: 10.2514/6.2018-1836

Balaji Kartikeyan Chandrasekaran,1 and James E Steck2 Wichita State University, Wichita, Kansas, 67220, USA

In the present age of increased demand for unmanned aerial vehicles, flapping wing unmanned aerial vehicle applications have become of interest, primarily because of their ability to fly silently and at lower speeds. This work explores new territory through the development of an adaptive flight controller for a bird-like flapping wing aircraft, using modified strip theory1 to model the aircraft’s aerodynamics and Newtonian equations of motion for the flight dynamics developed by Rashid2. The aircraft model is validated using existing data from the Slow Hawk Ornithopter given by zakaria3. The goal of this paper is to explore various adaptive flight control architectures, such as Model Reference Adaptive Control and Adaptive Neural Network Inverse Control, leading to an advanced controller to govern the longitudinal flight characteristics of the flapping wing aircraft. An approximate math model of the slow hawk ornithopter was developed in MATLAB/Simulink4. A Model Reference Adaptive Controller with Adaptive Bias Corrector was successfully able to adapt to uncertainties and improved the tracking performance compared to no adaptation. It was observed that with a B-Matrix failure the Adaptive controller was not able to reduce the tracking error to zero. The same observation was also made for system with adaptation and a PD controller. Another controller architecture in the form of Optimal Control Modification was utilized to control the system and the performance of different architectures were studied using the error metrics. OCM was able to adapt to the errors but higher learning rates exhibited a poor tracking performance and time delay margin. It was observed that OCM adaptation was able to successfully dampen the oscillations in system response.

I. Nomenclature ABC AFCS MST 𝑑⃗𝑏 𝛤

𝑖



𝜔 𝛽𝑜 ̅ 𝜃𝑤

1 2

= = = = = = = = =

Adaptive Bias Controller Advanced Flight Control System Modified Strip Theory 𝐷istance from body cg to point on root of wing aligned with wing cg Maximum flapping angle amplitude Cycle angle Flapping frequency Magnitude of dynamic Twist’s linear variation Mean pitch angle of chord

Graduate Research Assistant, Aerospace Engineering, and AIAA Student Member Professor, Aerospace Engineering, and AIAA Senior Member 1 American Institute of Aeronautics and Astronautics

Copyright © 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

II. Introduction


A. A Brief History One of the first known attempts of human flight was from Greek mythology in the form of story of Daedalus and Icarus (figure 1). They were a father-son duo who, in their attempt to escape from prison, built a pair of wings using twigs and wax. But Icarus flew too close to sun leading to melting of his wing and subsequent crash in sea and death by drowning. Vymaanika-Shaastra a 4th century BC text written by Maharishi Bhardwaj dealt with operation of ancient Vimana or flying crafts. One of which was called shakuna-vimana (figure 2) which was supposed to fly like a bird.

Figure 1. Daedalus and Icarus

Figure 2. Shakuna Vimana as described in Vymaanika-Shaastra

In modern history, Leonardo Da Vinci has been known to study the aerodynamics of these ornithopter even though he did not build any practical machines. The first documented flight of flapping-wing aircraft was achieved by Alphonse Penaud from France in 1874.

Figure 3. Powered Ornithopter of Lippisch5

Figure 4. Full-Scale Ornithopter “The Great Flapper” Built by Dr DeLaurier

Due to the idea that humans can only fly by imitating birds led to inventors like Lippisch, Emil Hartman in 1959 etc.,. To develop Human powered models. Lippisch5 designed a human powered ornithopter (figure 3) using the input given by Dr Brustmann, about the maximum power that can be developed by human arm and legs. He chose a flapping wing configuration due to its potentially greater efficiency compared to a fixed wing concept. He tested the concept with a pilot named Hans Werner Krause, whose goal was to reach a predetermined mark at about 250-300 meters from launch. The importance of this kind of flight was the fact that the acceleration (Thrust) and lift was provided by the flapping action of the wings alone. Etc. One of the most recent flights of an engine-powered ornithopter (figure 4) was designed and flown by DeLAurier6 from Institute of Aerospace Studies at University of Toronto . The “Great Flapper” was a full scale ornithopter powered by a 24hp engine and a flapping wing of 1.05 Hz with a span of 41.2 ft. The aircraft successfully lifted off the ground for a few seconds. 2 American Institute of Aeronautics and Astronautics


B. The Present The fixed wing aircraft has come a long way since wright brothers, motivated in part by World War I and World War II we graduated from simple wood and canvas construction to advanced metal construction and use of composites. With time even the speed, efficiency, carrying capability and safety also improved leading to present day passenger aviation. The fixed wing UAV has played an important and decisive role in war, like Iraq war, and in fight against terrorism. Civilian versions of these UAVs (both fixed wing and rotary wing) are nowadays utilized in agriculture, off-shore oil platforms, access tall cellphone towers and high placed power lines. Amid all this success with fixed wing aircraft, the idea of a flapping wing aircraft was forgotten until the advent of Unmanned Aerial Vehicles (UAV) in recent times. The research in this field led to design and development of various ornithopters which attempted to bio-mimic birds and insects. Delfly7, is a 3.07 g micro-ornithopter which has an on-board camera to enable a vision-based navigation. Nano hummingbird8, was a 19g ornithopter designed at Aerovironment as a part of the Defence Advanced Research Projects Agency’s (DARPA) requirement to develop a small, biologically inspired UAV that can sustain hover and fly forward at speeds of about 10 m/s. One of the widely used model9 is the “Slow Hawk Ornithopter “ developed by Sean Kinkade, which is also utilized in this work. A modified version of this aircraft was developed at University of Maryland Morpheus Lab known as Odyssey which is presently being used as a test bed for testing passive wing morphing10. Chen11 designed a unique butterfly ornithopter. C. Bird Flight The fundamental physics of flight of birds is very similar to that of aircraft, i.e., we need Lift to support weight and Thrust to overcome drag. The only difference is that instead of separating the lift and thrust producing mechanisms, a bird produces both by a single flap of wing. As shown in figure 5, Birds produce their Lift and Thrust by twisting their wing forward in down stroke and tilting it backward during upstroke. During down stroke it produces positive lift and Thrust but the upstroke produces reduced lift and possibly negative thrust. Different flapping species of animals can be distinguished based on wing movement12, which also affects the shape of their wings. Chatterjee13 studied the aerodynamics of Argentavis, one of the largest flying birds ever thought to exist based on fossils found in central and northwestern Argentina, and also its flight performance using computer simulations, which was an accomplished glider cruising at a speed of 67 kmph and a gliding angle close to 3 deg. Dial14 studied the evolution of avian flight by experimentally studying the wing strokes of ground birds. Hedenstrom15 studied the evolution of flight, the function of tails and also the ecological adaptations that the birds have to suit a flying lifestyle. Pennycuick16 calculated the power requirements for horizontal flight in pigeon. In a nutshell, a bird produces lift by moving the wing, thus producing a relative velocity over the airfoil. It produces thrust by active or passive morphing of the wing, which makes it act as a propeller. Tobalske17 studied the biological aspects of the bird flight which provided further insight into how the birds fly.

Figure 5. Bird Flapping Cycle D. Current Work 1. Ornithopter Modelling and Control Many models of ornithopter have been developed and were eventually controlled successfully. Shigeoka 18 developed an overall ornithopter transfer function model using experimental data acquired by strapping the ornithopter to a rail thus restricting its movement to forward horizontal direction. Acceleration and position sensor information were collected at different throttles. They subsequently developed a controller for its velocity and altitude. The velocity loop was controlled using a PI controller, and so was the altitude loop. Subsequently, a two-dimensional state-space model was also developed and its control and stability was discussed. 3 American Institute of Aeronautics and Astronautics


Julian19 modelled the flight dynamics of H2 Bird Ornithopter MAV shown in figure 6, a 13g MAV with a wingspan of 26.5 cm. using system identification techniques. They successfully developed a co-operative controller to help it traverse through a window with help of on-board sensors and a ground station based camera. Jackowski20 designed and constructed an autonomous ornithopter known as Phoenix as part of master’s thesis. The ornithopter was passively stable in other degrees of freedom but it was found difficult to control in pitch. A PD controller was found to be sufficient for pitch control and a proportional control was used for approximate velocity control. Krashanitsa21 studied the off-the shelf Cybird P2 ornithopter by subjecting it to thorough flight testing and developing the flight dynamics model using these results. The aerodynamic model was developed by subjecting the model to wind tunnel testing. 2. Model Reference Adaptive Controller In the work presented here in this paper we use a Model reference Figure 6. H2 Bird Ornithopter Adaptive Controller. Model Reference Adaptive Control (MRAC) is an adaptive flight control architecture which makes a non-linear system follow desired dynamics even in the presence of modelling error and failures. Early MRAC methods were introduced in the 1980s. Neural networks were used to adjust for unmodelled system dynamics by Narendra and Annaswamy 22 and Lewis23. The MRAC was implemented for controlling aircraft by Rysdyk24. They combined the adaptive control techniques with a linear inverse aircraft controller. Since the system adapts to change, it was demonstrated that MRAC can be utilized to control aircraft that are in loss of control regime as shown by Bosworth25, or if the aircraft is damaged as shown by Nguyen26. The General Aviation Flight Lab at Wichita State University has been conducting research on Adaptive control systems since mid-1990s. An early decoupled control method for general aviation was developed by Duerksen27 as part of his research funded by NASA’s Advanced General Aviation Transportation Experiment (AGATE) program. He developed a longitudinal decoupled flight controller with fuzzy logic that commanded flight path angle and airspeed. Steck28 researched an adaptive control system with decoupled pilot commands and simulated with a general longitudinal delta wing model and included the pitch attitude and airspeed commands. This technique included a linear compensator and an Artificial Neural Network that was trained offline. Since 2001, WSU partnered with Beechcraft Corporation to further develop such Advanced Flight Control System (AFCS) algorithms for general aviation applications. For purposes of modeling, simulation and flight testing, the algorithms developed were tested to a Beechcraft CJ-144 Small Aircraft Transportation System (SATS) Fly-By-Wire test bed. Work by Pesonen29 Advanced the AFCS into MRAC dynamic inverse controller. The more advanced version of the controller called the Adaptive Biased Controller (ABC) was developed by Steck30. The ABC was an ANN that included only the bias term of the neural network. The present work also utilizes the same architecture to explore the ABC’s ability to adapt ornithopter flight dynamics and model errors. The same architecture or a modified form was utilized in ref [19, 20]. E. Scope of this paper One of the common characteristics among the above mentioned ornithopter projects and Han’s work 31 was that they concentrated on developing the flight models for the particular aircraft model they were working with. Here we develop a general Simulink model for an ornithopter that can be utilized to simulate and develop controllers. Also, one of the major requirements that was considered was that, considering the advances in the field of morphing wings and their control21, it should also include the ability of the bird-like ornithopter to morph its wing shape to have an agile flight and maneuverability. The aim of this work is to develop a general non-linear model for a bird-like flapping wing aircraft using appropriate model for aerodynamics, which includes ability to consider the shape of the wings, and flight dynamics, followed by selection of a suitable specific aircraft model with all the necessary inputs. After the selection of a suitable model with all the geometric and other inputs required for the aerodynamics and flight dynamics model, a model follower Adaptive Non-linear Dynamic inversion controller is designed. A simple ANN is used and then its effect is studied in a scenario of actuator failure and other scenarios. A model of the ornithopter aerodynamics will be developed in MATLAB/Simulink environment and will be coupled with a suitable flight dynamics model. This general model will be simulated with an aircraft model to understand the dynamics of the system and effects of various A-matrix and B-Matrix failures on the system. Then the Adaptive component will be added to the system to study whether it is able to adapt to this aircraft model with 4 American Institute of Aeronautics and Astronautics

oscillating states. Since the aircraft flight dynamics is similar to that of fixed wing aircraft, with which the adaptation has been successful in tracking the commands in presence of failures, we will explore whetehr the adaptive controller will be successful with ornithopter flight dynamics in improving its tracking performance during a commanded pitch rate doublet. An OCM based approach for fixed wing aircraft was also adapted to develop a controller for the given system and its performance will be compared with that of the controller in previous paragraph using the error metrics.


III. Aerodynamic Model Various researchers have used a wide range of aerodynamic models for flapping wing aircraft. Some used experimental methods like wind tunnel and motion capture techniques like Krashanitsa 21. Grauer32, 33, and Shigeoka18 used an experimental technique of system identification by visual tracking. Rose34 considered a combined approach of wind tunnel measurements and free flight measurements using sensors. While Kaviyarasu and Kumar35 utilized an available simulation platform like X-plane to calculate the aerodynamics. The third approach to development of an aerodynamic model is a theoretical one based on steady-state aerodynamic models like Rashid2, who uses a quasisteady state aerodynamics model and ignores the unsteady wake effects. A. Wing Aerodynamics Model The aerodynamics model utilized here in this paper was developed by DeLaurier1. The model presented unsteady aerodynamics for an ornithopter using modified strip theory (MST). This model has been used by Kim36,Han31, Chalia and Bharti37 and Beng38. The model breaks down the entire wing into small sections of finite width and then the Lift, Drag and Moment is calculated for each section, based on the flow properties locally along with the kinematic effects of the flapping action to get the local flow characteristics. These local aerodynamic properties are then integrated over the entire wing to obtain the total lift, thrust and power. 1. Program The programming of the aerodynamic model was done by Beng38 in MATLAB and we have modified the code by retaining aerodynamic parts but,the Graphical User Interface (GUI) and the optimization module has been removed, to suit the needs of the present work The following assumptions was made by Beng during the programming

1. Negative-𝛼 ′ stalling does not occur. 2. Dynamic stall criterion is not incorporated. 3. The crossflow drag coefficient , (𝐶𝑑 )𝑐𝑓 was chosen to be that for a high-AR flat plate, given by Hoerner39 as 1.98. 4. The texture of wing’s surface is assumed to be such as to produce a full chord turbulent boundary layer. Thus, the friction drag coefficient, (𝐶𝑑 )𝑓 , obtained from Hoerner40 is given by (𝐶𝑑 )𝑓 =

0.89 [log(𝑅𝑒 )]2.58

(1)

B. Tail Aerodynamics Model The tail module is adopted from the work done by Grauer32. The model was developed using wind tunnel data. The values of these co-efficients are calculated based on the equations below: (2) 𝐶𝑥 = 𝐶𝑥𝑜 + 𝐶𝑥 2 𝛼 2 𝛼

𝐶𝑦 = 𝐶𝑦𝛼𝛽 𝛼𝛽

(3)

𝐶𝑧 = 𝐶𝑧𝑜 + 𝐶𝑧𝛼 𝛼 + 𝐶𝑧𝛽 𝛽

(4)

𝐶𝑙 = 𝐶𝑙𝛽 𝛽 + 𝐶𝑙𝛼𝛽 𝛼𝛽

(5)

𝐶𝑚 = 𝐶𝑚𝑜 + 𝐶𝑚 2 𝛽 2 + 𝐶𝑚𝛼 𝛼 + 𝐶𝑚𝛽 𝛽 + 𝐶𝑚𝛼𝑉 𝛼𝑉

(6)

𝛽

5 American Institute of Aeronautics and Astronautics

𝐶𝑛 = 𝐶𝑛𝛽 𝛽

(7)

In order to include the control variables, 𝑡𝑝𝑎 = 𝑇𝑎𝑖𝑙 𝑃𝑖𝑡𝑐ℎ 𝐴𝑛𝑔𝑙𝑒 and 𝑡𝑟𝑎 = 𝑇𝑎𝑖𝑙 𝑅𝑜𝑙𝑙 𝑎𝑛𝑔𝑙𝑒, certain changes were made to equations (2)-(7). The pitch control variable , 𝑡𝑝𝑎 , was added to α of the tail. Hence, the angle of attack of the tail is sum of body angle of attack and 𝑡𝑝𝑎 . The lateral control variable 𝑡𝑟𝑎 , is incorporated in a slightly different way. The lateral control of the ornithopter is achieved by re-positioning the angle at which the forces act so that some of the lift generated by the tail is utilized for generating a small side force. Thus, the aerodynamic forces X, Y and Z of that tail are rotated by roll angle of 𝑡𝑟𝑎 . Thus the new forces considering the roll angle is given by:


F = [1 0 0; 0 cos(t_ra) sin(t_ra); 0 − sin(t_ra) cos(t_ra)] ∗ [X; Y; Z]

(8)

Also, there is small change made in the Yawing moment equation, where a moment arm is added to add the effect of side-force produced by re-directing the lift of tail. (9) 𝑁 = 𝐶𝑛 𝑄𝑆𝑡 𝑏𝑡 + 2 ∗ 𝐹(2) Where, 2 = longitudinal moment arm in ft 𝐹(2) = Control force in y-direction in body axis

IV.

Flight Dynamics

The model presented here can be called the 2-panel method, figure (7). This method assumes that an ornithopter can be broken down into body, including tail and fin (if any), and 2-wing panels. The forces and moments produced by wings are coupled to body through the reactionary forces at the hinge points where they are attached to the body as shown in figure (8)

Figure 7. 2-Panel Method

Figure 8. Position of Body and Hinge Joints with Forces and Moments

The equations of motion are written down separately for body and each of the wings and then they are connected through the kinematics equations. The final equations are given as a set of 18 Linear equations with 18 unknowns, namely the states U, V, W, P, Q, R and the reaction forces and moments at the hinge joints between each of the wings and body. The present work converts them into 6-DOF equations which then becomes easier to model in MATLAB/Simulink. The final expressions in vector form is given in equations (10) and (11). These equations are from the author’s thesis41 and more details can be found there.


Force vector Equation

⃗⃗̇ + 𝑀 𝜔̇⃗⃗ 𝑋(𝑑⃗ −𝐷 ⃗⃗𝑤1 ) + 𝑀𝑤2 𝜔̇⃗⃗𝑏 𝑋(𝑑⃗𝑏 −𝐷 ⃗⃗𝑤2 ) = (𝑀𝑏 + 𝑀𝑤1 + 𝑀𝑤2 )𝑉 𝑏 𝑤1 𝑏 𝑏𝑤1 𝑤2 ⃗⃗𝑏 ) 𝐹⃗𝑏 + 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑤1 + 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑤2 + 𝑀𝑏 (𝜔 ⃗⃗𝑏 𝑋𝑉 𝑏

𝑤1

(10)

𝑤2

⃗⃗𝑏 + (𝜔 ⃗⃗𝑤1 )) − 𝑀𝑤1 [(𝜔 ⃗⃗𝑏 𝑋(𝜔 ⃗⃗𝑏 𝑋𝑑⃗𝑏𝑤1 )) + 𝜔 ⃗⃗𝑏 𝑋𝑉 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑤1 )) + 𝜔̇⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑤1 ) + (𝜔 ⃗⃗𝑤1 )) + (2𝜔 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑐1 𝑋𝜔 ⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑏 + 𝜔 ⃗⃗𝑤1 ) + 𝜔 ⃗⃗𝑤1 ))] +𝜔 ⃗⃗𝑤1 𝑋(𝑉 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑏 𝑋𝑑⃗𝑏 + 𝜔 ⃗⃗𝑐1 𝑋(−𝐷 𝑤1

⃗⃗𝑏 + (𝜔 ⃗⃗𝑤2 )) − 𝑀𝑤2 [(𝜔 ⃗⃗𝑏 𝑋(𝜔 ⃗⃗𝑏 𝑋𝑑⃗𝑏𝑤2 )) + 𝜔 ⃗⃗𝑏 𝑋𝑉 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑤2 )) + 𝜔̇⃗⃗𝑐2 𝑋(−𝐷 ⃗⃗𝑤2 ) + (𝜔 ⃗⃗𝑤2 )) + (2𝜔 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑐2 𝑋(−𝐷 ⃗⃗𝑐2 𝑋𝜔 ⃗⃗𝑐2 𝑋(−𝐷 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ +𝜔 ⃗⃗𝑤2 𝑋(𝑉𝑏 + 𝜔 ⃗⃗𝑏 𝑋(−𝐷𝑤2 ) + 𝜔 ⃗⃗𝑏 𝑋𝑑𝑏 + 𝜔 ⃗⃗𝑐2 𝑋(−𝐷𝑤2 ))]


𝑤2

Moment vector Equation

⃗⃗̇ + 𝑀 𝜔̇⃗⃗ 𝑋(𝑑⃗ ⃗⃗𝑤1 )𝑋 (𝑀𝑤1 𝑉 ⃗⃗ 𝐼𝑏 𝜔̇⃗⃗𝑏 + 𝐼𝑤2 𝜔̇⃗⃗𝑏 + 𝐼𝑤1 𝜔̇⃗⃗𝑏 + (𝑑⃗𝑏𝑤1 − 𝐷 𝑏 𝑤1 𝑏 𝑏𝑤1 − 𝐷𝑤1 ))

(11)

⃗⃗̇ + 𝑀 𝜔̇⃗⃗ 𝑋(𝑑⃗ ⃗⃗𝑤2 )𝑋 (𝑀𝑤2 𝑉 ⃗⃗ + (𝑑⃗𝑏𝑤2 − 𝐷 𝑏 𝑤2 𝑏 𝑏𝑤2 − 𝐷𝑤2 )) ⃗⃗⃗𝑎𝑒𝑟𝑜 + 𝑀 ⃗⃗⃗𝑎𝑒𝑟𝑜 + 𝑀 ⃗⃗⃗𝑎𝑒𝑟𝑜 − 𝐼𝑤1 𝜔̇⃗⃗𝑐1 − 𝐼𝑤2 𝜔̇⃗⃗𝑐2 − 𝜔 = 𝑀 ⃗⃗𝑏 𝑋𝐼𝑏 𝜔 ⃗⃗𝑏 𝑤1 𝑤2 𝑏 ̇ ̇ −𝜔 ⃗⃗𝑤1 𝑋𝐼𝑤1 𝜔 ⃗⃗𝑤1 − 𝜔 ⃗⃗𝑤2 𝑋𝐼𝑤2 𝜔 ⃗⃗𝑤2 − 𝐼𝑤1 𝜔 ⃗⃗𝑤1 − 𝐼𝑤2 𝜔 ⃗⃗𝑤2 ⃗⃗𝑤1 )𝑋 (𝑀𝑤1 [(𝜔 ⃗⃗𝑏 + (𝑑⃗𝑏 −𝐷 ⃗⃗𝑏 𝑋(𝜔 ⃗⃗𝑏 𝑋𝑑⃗𝑏 )) + 𝜔 ⃗⃗𝑏 𝑋𝑉 𝑤1

𝑤1

⃗⃗𝑤1 )) + (2𝜔 ⃗⃗𝑤1 )) + 𝜔̇⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑤1 ) + (𝜔 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑤1 )) + 𝜔 ⃗⃗𝑏 + 𝜔 ⃗⃗𝑤1 ) + 𝜔 + (𝜔 ⃗⃗𝑐1 𝑋𝜔 ⃗⃗𝑐1 𝑋(−𝐷 ⃗⃗𝑤1 𝑋(𝑉 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑏 𝑋𝑑⃗𝑏

𝑤1

⃗⃗𝑤1 ))] + 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑤1 ) + 𝜔 ⃗⃗𝑐1 𝑋(−𝐷 𝑤1 ⃗⃗𝑤2 )𝑋(𝑀𝑤2 [(𝜔 ⃗⃗𝑏 + (𝑑⃗𝑏𝑤2 − 𝐷 ⃗⃗𝑏 𝑋(𝜔 ⃗⃗𝑏 𝑋𝑑⃗𝑏𝑤2 )) + 𝜔 ⃗⃗𝑏 𝑋𝑉 ⃗⃗𝑤2 )) + (2𝜔 ⃗⃗𝑤2 )) + 𝜔̇⃗⃗𝑐2 𝑋(−𝐷 ⃗⃗𝑤2 ) + (𝜔 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑏 𝑋𝜔 ⃗⃗𝑐2 𝑋(−𝐷 ⃗⃗𝑤2 )) + 𝜔 ⃗⃗𝑏 + 𝜔 ⃗⃗𝑤2 ) + 𝜔 + (𝜔 ⃗⃗𝑐2 𝑋𝜔 ⃗⃗𝑐2 𝑋(−𝐷 ⃗⃗𝑤2 𝑋(𝑉 ⃗⃗𝑏 𝑋(−𝐷 ⃗⃗𝑏 𝑋𝑑⃗𝑏

𝑤2

⃗⃗𝑤2 ))] + 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑤2 + 𝜔 ⃗⃗𝑐2 𝑋(−𝐷 𝑤2

V.

Simulink Model and Aircraft Selection

A. Introduction There are various physical models that the researchers use for ornithopter research that can serve as an example for our model, but since majority of them are proprietary or the data available are not sufficient to complete the requirements of our sub-systems, our choice was very limited. One of the options was the “The Great Flapper” that was built by Dr DeLaurier. Even though the wing section details, like number of sections, chord length of each section, airfoil of each section etc.,.and mass details were available for building the aerodynamics model, necessary information like wing CG location, Moment of Inertia etc., were not available to complete the flight dynamics. The second option was the more common UAV “Slow Hawk Ornithopter” that was built by Sean Kinkade, which is very common among hobby enthusiasts. The geometric details of this selected aircraft are provided in the following subheadings. In this paper, the aerodynamics model developed by Dr DeLaurier is being used in combination with flight dynamics model developed by Rashid. The aircraft selected to be simulated is the “slow hawk ornithopter”. B. Aerodynamics Block As we can see in the figure (9), the aerodynamics block has the inputs as enumerated in table 1 below. Velocity and angle of attack are states from the flight dynamics model while the tail angles act as control variables.


OUTPUT S


INPUT S

Figure 9. Aerodynamic Block Showing Inputs and Outputs

Table 1. Aerodynamic Block Inputs Input Variable

Units

Frequency

Hz

𝑉𝑏

ft/s

Description The frequency with which the aircraft wings flap The velocity vector of aircraft in body axis Angle of attack of the aircraft body

alpha

rad

calculated as the inverse tangent of ratio of vertical velocity of the body to forward velocity of the body

𝑡𝑝𝑎

rad

The tail setting angle, also referred to as tail pitch angle. The tail roll angle, it is the angle by

𝑡𝑟𝑎

rad

which the tail will be rolled about the body x-axis for lateral axis control

The outputs from the aerodynamic model are the forces and moments, and includes other important variables that are required by flight dynamics block. The details of every output is presented in table (2) below:



Table 2 Aerodynamic Block Outputs Output Variable

Symbol

Units

Force-Body

𝐹⃗𝑏

lbf

Force-wing 1

𝐹⃗𝑤1

lbf

Force-wing 2

𝐹⃗𝑤2

lbf

Moments-body

⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀 𝑏

lbf-ft

The aerodynamic moments generated by body-tail combination

Moments-wing 1

⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀 𝑤1

lbf-ft

The aerodynamic moments generated by port (left) wing

Moments-wing 2

⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀 𝑤2

lbf-ft

The aerodynamic moments generated by starboard (right) wing

Omega_c1

𝜔 ⃗⃗𝑐1

rad/sec

The angular velocity of the port (left) wing due to flapping

Omega_c2

𝜔 ⃗⃗𝑐2

rad/sec

The angular velocity of the starboard (right) wing due to flapping

Omega_c1_dot

𝜔̇⃗⃗𝑐1

rad/sec2

The angular acceleration of the port(left) wing

Omega_c2_dot

𝜔̇⃗⃗𝑐2

rad/sec2

The angular acceleration of the starboard (right) wing

Gamma

Description The aerodynamic forces in all three directions developed by body and tail combined The aerodynamic forces in all three directions developed by port (left) wing The aerodynamic forces in all three directions developed by starboard (right) wing

Flapping angle of the wing which is a function of cycle angle, ∅ and

rad

the flapping amplitude in the aerodynamic model

The wing aerodynamic model is introduced in the SIMULINK model as an s-function. The inputs to this s-function is enumerated in table (3). Table 3. Inputs to Aerodynamic S-Function Variable Alpha0In EtaS Cmac MaxAlphaStallIn Cs ThetaWIn

Units rad No Dim No Dim rad ft rad

Description Zero Lift Angle Leading edge suction efficiency Co-efficient of moment about aerodynamic center of wing Maximum stall angle of the airfoil Chord length of each section of wing Mean pitch angle of chord with respect to pitch axis

Beta0

Symbol 𝛼𝑜 𝜂𝑠 𝐶𝑚𝑎𝑐 (𝛼𝑠𝑡𝑎𝑙𝑙 )𝑚𝑎𝑥 cs ̅ 𝜃𝑤 𝛽𝑜

rad

Magnitude of dynamic twist’s linear variation

Ys Dy

𝑦 dy

ft ft

Co-ordinate along semi-span Width of each section of a wing

C. Aircraft Parameters-Aerodynamics 1. Wings As mentioned in the introduction, the aircraft selected for simulation in this thesis is the “Slow Hawk Ornithopter” built by Kinkade9, 42. Various properties and geometric dimensions required for the aerodynamics model developed in previous sections are available from the work of Zakaria3. The values of the variables like flapping angle magnitude, pitch angle of flapping axis and magnitude of dynamics twist’s linear variation were selected , though different from ones chosen by Zakaria3, in such a way that the lift generated by the wings matched the weight of the aircraft at the given speed. Their values are given in table 4.


Table 4. Values of Initialized Parameters Variable

Description

Value

Γ

Flapping angle Magnitude

35 deg

𝜃𝑎̅

Pitch angle of flapping axes

5.2 deg

𝛽𝑜

Magnitude of dynamics twist’s linear variation

0.5 deg

𝒇 (Hz)

𝑼(ft/sec)

𝜞(deg)

̅𝒂 (deg) 𝜽

𝜷𝒐 (deg/ft)

Thrust(lbs)

3.24

15.65

35

5

35

0.9259415

2

1.5

Lift(in lbs)

Lift (lbs) 0.97

The aerodynamic block was populated and run and the calculated Lift, Thrust and Moments of the wing are shown below in figures 10, 11 and 12. Since optimization was not the goal of this work, we chose a set of values for various design variables to generate enough lift whose magnitude is equal and opposite to the weight of the aircraft. The values of these variables are given in table 5.

2.5

1

0.5

0

-0.5

0

1

2

3

4 5 6 Time (in secs)

7

8

9

10

Figure 10. Lift Generated by Wings

0.9

1.5

0.8

1 Pitching Moment (in lbs-ft)

0.7

0.5

Thrust (in lbs)


Table 5. Input Parameters

0

-0.5

0.6 0.5 0.4 0.3 0.2 0.1

-1 0

-1.5

0

1

2

3


7

8

9

0

1

2

3


7

8

9

10

10

Figure 11. Total Thrust Generated by Wings

Figure 12. Total Pitching moment Generated by Wings



1.1 Tail

Figure 13. Tail Aerodynamic Co-Efficients from Grauer In the model considered in this work, the signs for the values of 𝐶𝑚𝛼 𝑎𝑛𝑑 𝐶𝑚𝛼𝑉 has been changed in order to have a stable aircraft model after substituting the values for other variables. Various aerodynamic co-efficients are given below from Grauer32. The forces and moments generated by tail in a trim condition are given in table 6. Table 6. Tail Forces and Moments Values Forces Drag Lift Side Force Pitching Moment

Value -0.0401 lbs 0.0385 lbs 0 lbs -0.4892 lbs-ft

D. Flight Dynamics Block The equations (10) and (11) were coded in SIMULINK. Figure 15 shows the block.

INPUT S

OUTPUTS

Figure 14. Flight Dynamics Block Showing Inputs and Outputs 11 American Institute of Aeronautics and Astronautics

The inputs and outputs are explained in the following tables. Table 7. Flight Dynamics Block Inputs Block input Force_body Force_wing 1 Force_wing 2 Moments_body Moments_wing 1 Downloaded by WICHITA STATE UNIVERSITY on February 7, 2018 | http://arc.aiaa.org | DOI: 10.2514/6.2018-1836

Moments_wing 2 Omega_c1 Omega_c2 Omega_c1_dot Omega_c2_dot dI_w1 dI_w2

Equivalent variable in equations 10 and 11 𝐹⃗𝑏 + 𝐹⃗𝑎𝑒𝑟𝑜 𝑏

𝐹⃗𝑤1 + 𝐹⃗𝑎𝑒𝑟𝑜𝑤1 𝐹⃗𝑎𝑒𝑟𝑜 + 𝐹⃗𝑤2 𝑤2

⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀 𝑏 ⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀

𝑤1

⃗⃗⃗𝑎𝑒𝑟𝑜 𝑀 𝑤2 𝜔 ⃗⃗𝑐1 𝜔 ⃗⃗𝑐2 𝜔̇⃗⃗𝑐1 𝜔̇⃗⃗𝑐2 ̇ 𝐼𝑤1 ̇ 𝐼𝑤2

Table 8. Flight Dynamics Block Outputs Block output Vb_dot Wb_dot Euler Ve Xe V_b W_b Alpha Out 1

Out 2

Equivalent variable in equations 70 and 71 ⃗⃗̇ 𝑉 𝑏 𝜔̇⃗⃗𝑏 ∅, 𝜃, 𝛹 𝑥̇ , 𝑦̇ , 𝑧̇ (From equation 58 and 59) x,y,z ⃗⃗𝑏 𝑉 𝜔 ⃗⃗𝑏 𝛼 A port to tap into the current moment generated by wing 1 to be forwarded to Dynamic inverse controller A port to tap into the current moment generated by wing 2 to be forwarded to Dynamic inverse controller

̇ And 𝐼𝑤2 ̇ are calculated because the flapping wing changes the values due to the flapping motion. A 𝐼𝑤1 generalized equation was developed based on the one derived by Rashid2 in appendix E. This equation was then coded as an S-function in MATLAB. The model also consists of an S-Function in the end to calculate the linear and angular accelerations. The output from this S-function is then further integrated to get the linear velocity and angular velocity in body-axis. These velocities are then used for further processing to solve for Euler angles. These angles are further utilized to solve for trajectory in the inertial frame of reference. All of these processing utilizes the standard aircraft equations from Roskam43.


E. Aircraft-Inputs Flight Dynamics There are two more main sets of inputs that are needed for successful execution of the overall SIMULINK model. The first one is the overall global variables like Mass of various components, density value, etc. Secondly, the inputs required to initiate the Flight dynamics model. Table 9. Global Inputs


Variable Mass_body Mass_wing 1 Mass_wing 2 Inertia_wing 1

Value 0.75 lbs 0.0912 lbs 0.0912 lbs [0.0217147 0 0; 0 0.0100256 0; 0 0 0.03173936] [0.0217147 0 0; 0 0.0100256 0; 0 0 0.03173936] [0.0026713267 0 0; 0 0.09015876 0; 0 0 0.08873731];

Inertia wing 2

Inertia_body

Table 10. Miscellaneous Inputs Variable ⃗⃗𝑤1 𝐷 ⃗⃗𝑤2 𝐷 𝑑⃗𝑤1 𝑑⃗𝑤2

VI.

Value (ft) [0 -0.1 0]’ [0 0.1 0]’ [0 0 0]’ [0 0 0]’

Controller Design

A. Controller Architectures Three kinds of adaptive control architectures are utilized here; Adaptive Bias Controller (ABC) with PI Controller, ABC with PD Controller and OCM-Linear model. Each of these architectures differ either in their linear controller or the adaptive architecture.

Figure 15. General Overview of Controller architecture



1. Model Follower The first component of the controller is the model follower shown in figure (16). This model depicts the desired dynamics of the system. It is adapted from the work by Nguyen44

Figure 16. Model Follower The reference model is a first-order system calculating 𝜔𝑚 for the input vector of commanded aircraft rotational rates, 𝜔𝑐 and is given by Nguyen44 (12) 𝜔̇ 𝑚 + 𝜔𝑛 𝜔𝑚 = 𝜔𝑛 𝜔𝑐 Where 𝜔𝑛 is the matrix of natural frequencies of each axis along the diagonal. For longitudinal controller, equation (12) is applied only to the pitch rate command,𝑞𝑚 , which gives us the following equation 𝑞̇ 𝑚 + 𝜔𝑛 𝑞𝑚 = 𝜔𝑛 𝑞𝑐𝑜𝑚 Laplace Transforming, we can see that the above equation is a first-order system with a time constant of

(13) 1 𝜔𝑛

. To

design this controller the preferred parameter is the rise time of the system,𝑇𝑟 as 2.2 (14) 𝜔𝑛 = 𝑇𝑟 , In order to avoid transmitting a noisy error signal, the derivative portion is taken directly from the model as 𝑞̇ 𝑚 . This is then added to the output of PI controller and the adaptive signal to form 𝑞̇ 𝑐𝑜𝑚 . The rise time for the PI controller was chosen to be 0.5 secs considering the type of aircraft and the requirement to catch up to commanded signal quickly for agility. Hence, the model follower’s rise time was fixed at 0.275 secs so that it catches up with command sooner than the controller. 2. Inverse Controller The inverse controller used in this control architecture was derived by inverting equation 11 to solve for t_pi (input). The input to the controller is the commanded pitch acceleration 𝑞̇𝑐𝑜𝑚 . The inverse controller generate the required control values of the tail setting angle, t_pi to achieve the desired accelerations. Inversion of equation 11 leads to the equation below, after substituting the values of known variables: 𝑞̇ = 0.0016812 𝑉 2 [−0.3486 − 3.3182 𝛼 + 0.0975 𝛽 − 0.4184 𝛽 2 + 0.3053 𝛼𝑉] + 𝑀𝑤1 + 𝑀𝑤2 − 𝑝𝑟(𝐼𝑥𝑥𝑏 − 𝐼𝑧𝑧𝑏 ) − 𝑟(𝑝 − 𝜔𝑐1 )(𝐼𝑥𝑥𝑤1 − 𝐼𝑧𝑧𝑤1 ) − 𝑟(𝑝 − 𝜔𝑐2 )(𝐼𝑥𝑥𝑤2 − 𝐼𝑧𝑧𝑤2 ) ̇ ̇ − 𝑞(𝐼𝑦𝑦 + 𝐼𝑦𝑦 ) 𝑤1 𝑤2

(15)

3. Adaptive Bias Correction The adaption used here is known as the Adaptive Bias Controller. It is the key component in the Model Reference Adaptive Controller. The adaptive element adjusts the inputs to the inverse controller by accounting for the modelling error between aircraft and the inverse controller, which is a very important requirement since the dynamics of an ornithopter is still uncertain. The Adaptive Bias Controller was developed at Wichita State University as a simple adaptive element. The ABC adaptation uses a simple bias neural network element that parametrizes the modelling or tracking-error to form the corrective signal of 𝑞̇ 𝑎𝑑𝑑 such that 14 American Institute of Aeronautics and Astronautics

𝑞̇ 𝑎𝑑𝑑

𝑛𝑒𝑤

= 𝑞̇ 𝑎𝑑𝑑 𝑜𝑙𝑑 + 𝜂𝑒(𝑡)

(16)

Where, 𝜂 is the learning rate and 𝑒(𝑡) is the tracking error. 4. Proportional-Integral (PI) Controller The fourth component of ABC-PI controller is the PI controller. This was also taken from work by Nguyen44. In a linear analysis assuming the inverse controller and the aircraft perfectly cancel each other to become an integrator, using a gain other than unity for 𝐾𝑝 will result in 𝑞 not exactly following, 𝑞𝑚 . The PI controller in the frequency domain is represented as 𝐾 (17) 𝑞̇ = ( 𝑖 + 𝐾 )𝑞 𝑑𝑒𝑠

𝑠

𝑝

𝑒


Where, 𝑞𝑒 = 𝑞𝑚 − 𝑞. Nguyen44 recommend relating the gains to the desired dynamics of the system specifying a damping ratio,𝜁, and natural frequency, 𝜔𝑛 , in the case of the longitudinal controller 𝐾𝑝 = 2𝜁𝜔𝑛 𝐾𝑖 = 𝜔𝑛 2

(18)

The rise time for the system was chosen to be 0.5 secs for the reason as mentioned in model follower, with a damping ratio of 0.9. 5. Proportional-Derivative (PD) Controller One of the changes in the architecture that has been studied here is replacing the PI controller with a PD Controller in fig (17) . The PD controller in time domain is presented as follows (19) 𝑞̇ 𝑑𝑒𝑠 = (𝐾𝑑 𝑠 + 𝐾𝑝 )𝑞𝑒

Figure 17. PD Controller

6. Optimal Control Modification Optimal control modification was introduced by Nguyen45 to achieve fast adaptation for MRAC. Reed46 utilized this concept to develop a OCM architecture and compared its performance with other architectures. The OCM derives its name from the optimal control techniques used for weight updates. The variation used here is using the adaptive parameterization given below; 𝑞̇ 𝑎𝑑𝑑 = 𝛩𝑇 ∅ = 𝛩𝑇 [𝑞 𝜃 𝛼]𝑇

VII.

(20)

Major model differences

A. Flight Dynamics Model During early design stages it was found that the tail size and its aerodynamic co-efficients caused it to oscillate with very large amplitude during trim conditions. The tail oscillates in this model in order to counteract the pitching moment generated by the model, which is a result of the wing flapping. Hence, in order to reduce these oscillations, thus reducing the actuator effort, the tail was assumed to be bigger than the original design and placed such that the effectiveness of the tail is scaled up by a factor of five. Also, the trim value of the tail setting angle was found to be beyond normal operational limits. Hence, in order to bring this value within acceptable, practical levels the 𝐶𝑚𝑜 of the tail was scaled up by a factor of three.


B. Lateral Control Loop During initial trials it was observed that the flapping motions of the wings made the aircraft drift slightly sideways, a small yawing and rolling moment, which was seen to affect the longitudinal variables. Since lateral control was out of scope of the current work, a Lat-Dir PI controller was chosen and was tuned automatically using MATLAB/Simulink.


VIII.

Results

The results presented are the time response tracking of the final design gains for the controller. The time response of the controller to a pitch rate doublet will be shown for the pitch rate, angle of attack, airspeed, aircraft pitch angle along with the control input, and tail setting angle. The results compare the performance of the system with and without the adaptation. Stepanyan47 introduced the concept of error metrics which can be used to verify and validate various adaptive systems. Reed46 utilized these methods to tune the gains for the ABC and OCM controllers. We will be utilizing the transient performance metric , described in Section II.B(1) by stepanyan 47. This metric will be used to measure the oscillations of the adaptive system with respect to the model follower and will serve as the method to compare performance of different architectures in different conditions. The metric is the integration of the second-norm of the tracking error for a pitch doublet from t = 20 secs to 35 secs ‖𝑞𝑚 (𝑡) − 𝑞(𝑡)‖𝐿2 (21) 𝑀= ‖𝑞𝑚 (𝑡)‖𝐿2 This chapter has four sections. The first section presents the results of the PI control architecture described in the previous sections for A-Matrix errors in the aircraft model. The second section presents the results of the control architecture with a PD element in place of the PI controller along with the same modelling errors as introduced for the PI controller. The third section consists of the response from OCM modification. The fourth section presents the result of performance of the above mentioned architectures in case of the tail actuator failure, a B-Matrix error, which is assumed to reduce the effectiveness of the tail by half. A. PI Controller The gains of this controller rare given in table 10. With no delay errors the time response of the controller to the pitch Table 11. PI Controller Gains rate doublet is shown in figure 18 and 19. This controller was able to track the commanded value of pitch rate. The oscillations observed in Variable Value the figures are due to the flapping of the wings. The values plotted here 𝐾𝑝 10.38 are the forward velocity u, angle of attack α, pitch angle θ and q. The second case presents the performance of the PI controller when 33.25 𝐾𝑖 an error is introduced in the model by reducing 0.02 𝜂 the 𝐶𝑚𝛼 to zero in the aircraft model, but Inverse Controller uses the original value. The controller was unable to exactly follow the commanded pitch rate as we can see in figure 20. The ANN was introduced again and it can be observed that the ANN adapted to the modelling error and reduced the tracking error as observed from the 20-25 sec mark in the q plot of fig 21


u

alpha

15.5

15.3

alpha (rad)

forward Velocity (ft/sec)

0.05 15.4

15.2 15.1

-0.05

15 -0.1

14.9 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

Theta

30

q 0.15 actual model

0.1

q (rad/sec)

0.15

0.1

0.05 0 -0.05

0.05 -0.1 0 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

Figure 18. PI Controller-Original System

Input 0.8

0.6

0.4

Input (rad)

Theta (rad)


0

0.2

0

-0.2

-0.4 10

15

20

25

30

Time (secs)

Figure 19. Input


30

0.05 15.4

15.2

0

-0.05

15 -0.1 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

Theta

30

q 0.15 actual model

0.1

q (rad/sec)

Theta (rad)

0.15

0.1

0.05

0.05 0 -0.05 -0.1

0 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

30

Figure 20. PI Controller +Zero Cm-alpha

u

alpha 0.05

15.4 15.3

alpha (rad)


15.5

15.2 15.1

0

-0.05

15 -0.1

14.9 10

15

20 25 Time (secs)

30

10

15

Theta

20 25 Time (secs)

30

q 0.15 actual model

0.1

q (rad/sec)

0.15

Theta (rad)


alpha

alpha (rad)


u

0.1

0.05

0.05 0 -0.05 -0.1

0 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

Figure 21. PI Controller +Zero Cm-alpha+ ANN


30

The third case is to show the performance of the PI controller when an error is introduced in the model by having a value of 𝐶𝑚𝛼 in the aircraft model that is unstable with a magnitude of 100% of its stable value, but Inverse Controller used the original value. Consistent with the previous results the system is not able to track the commanded q from 2025 sec mark. The ANN was able to adapt to this modelling error and the results are presented in fig 23. Also, consistent with its previous trend seen above, ANN enabled the system to track the commanded q and the it also improved upon its error at the 22-25 sec mark.

15.4 alpha (rad)


alpha 0.05

15.2

0

-0.05

15 -0.1 10

15

20 25 Time (secs)

30

10

15

Theta

20 25 Time (secs)

30

q 0.15 actual model

0.1 q (rad/sec)

Theta (rad)

0.15

0.1

0.05

0.05 0 -0.05 -0.1

0 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

30

Figure 22. PI Controller + case 2 unstable value of Cm-alpha

alpha 0.05

15.4

alpha (rad)


u

15.2

15

0 -0.05 -0.1

10

15

20 25 Time (secs)

30

10

15

Theta

20 25 Time (secs)

30

q 0.15 actual model

0.1

q (rad/sec)

0.15

Theta (rad)


u

0.1 0.05

0.05 0 -0.05 -0.1

0 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

30

Figure 23. PI Controller + case 2 unstable value of Cm-alpha+ANN


B. PD Controller The gains of the controller rare given in table 11. With no delay errors the time response of the controller to the pitch rate doublet is shown in figure 24 and 25. This controller was able Table 12. PD Controller Gains to track the commanded value of pitch rate. The oscillations Variable Value observed in the figures are due to the flapping of the wings. The 𝐾𝑝 10 values plotted here are the forward velocity u, angle of attack α, pitch angle θ and the pitch rate q. 1.27 𝐾𝐷 0.02 𝜂

0.05

alpha (rad)


alpha

15.4 15.3 15.2 15.1

0 -0.05

15 10

-0.1 15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

Theta

30

q 0.15

0.15

0.1

0.1

0.05

q (rad/sec)

Theta (rad)

0.05

10

actual model

0 -0.05 -0.1

0 15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

30

Figure 24. PD Controller-Original System Input 0.8

0.6

0.4

Input (rad)


u 15.5

0.2

0

-0.2

10

15

20

25

30

Time (secs)

Figure 25. PD Controller Input The second case is to show the performance of the PD controller when an error is introduced in the model by reducing the 𝐶𝑚𝛼 to zero in the aircraft model, but the Inverse Controller uses the original value. The controller was again unable to exactly follow the commanded pitch rate as we can see in figure 26. The ANN was introduced again and it can be observed that the ANN adapted to the modelling error and reduced the tracking error as observed from the 20-25 sec mark in the q plot of fig 27. The system responds like a first order system but it can be observed at the 20 American Institute of Aeronautics and Astronautics

22 sec mark and 25 sec mark that the system’s rise time is longer than the first case leading to an error from the pitch rate model.

15.5 0.05 15.4 15.3 15.2

0 -0.05

15.1 10

15

20 25 Time (secs) Theta

30

10

15

20 25 Time (secs) q

30

0.15

q (rad/sec)

Theta (rad)

actual model

0.1

0.1 0.05 0

0.05 0 -0.05 -0.1

-0.05 10

15

20 25 Time (secs)

30

10

15

20 25 Time (secs)

30

Figure 26. PD Controller +Zero Cm-alpha

u

alpha

15.4 15.3

alpha (rad)


0.05

15.2 15.1

0

-0.05

15 14.9 10

-0.1 15

20 25 Time (secs)

30

10

15

Theta

20 25 Time (secs)

30

q 0.15

0.15

actual model

0.1

q (rad/sec)

Theta (rad)


alpha

alpha (rad)


u

0.1

0.05

0.05 0 -0.05 -0.1

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Figure 27. PD Controller +Zero Cm-alpha+ ANN


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The third case is to show the performance of the PD controller when an error is introduced in the model by having a value of 𝐶𝑚𝛼 in the aircraft model that is unstable with a magnitude of 100% of its stable value, but the Inverse Controller uses the original version. Consistent with the previous results the system is not able to track the commanded q from 20-25 sec mark. The ANN was able to adapt to this modelling error and the results are presented in fig 29. Also, consistent with its previous trend seen above, ANN enabled the system to track the commanded q but the rise time is seen to have increased.

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Figure 28. PD Controller + case 2 unstable value of Cm-alpha

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Figure 29. PD Controller + case 2 unstable value of Cm-alpha+ANN



C. OCM Architecture The OCM architecture that was developed for a fixed wing aircraft was adopted to the present work and its response to various errors were studied by varying its learning rates. A time delay error metric study was also carried out to identify trends in robustness of the controller with a particular set of learning rates. Three learning rates are used here 250, 10000 and 50000. Each of these controllers were coupled with original system and their responses were recorded. Subsequently, modelling error in the form of unstable 𝐶𝑚𝛼 , similar to third case of the PI and PD controllers above were introduced and the responses were recorded and presented here. Finally, the reduced tail effectiveness scenario was also introduced and the response is of each iteration of the controller is presented in the following section. Figures (30) and (31) present the response of the controller with learning rate 250. It can be observed that the OCM has no discernible effects on the system response as compared to a PI controller response in figures (18) and (22).

Figure 30. OCM-250 + Original system

Figure 31. OCM-250 + case 2 unstable value of Cm-alpha



Figures (32) and (33) present the response of the controller with learning rate 10000. It can be observed that the OCM has no discernible effects on the system response as compared to a PI controller (figure (18)) response for the original system with no uncertainty but with modelling error, the OCM performs well as compared to PI controller (figure(22)). Another feature to notice is that OCM adaptation dampens out the system oscillations, observed in the q plot of fig 32, in system response while the ABC controller in figure 23 could not.





Figures (34) and (35) present the response of the controller with learning rate 50000. It can be observed that this iteration of the controller is has a similar response as the original system with no uncertainties but has much better improved tracking performance for the modelling error case. The improved damping of the system response is also observed here.




D. Reduced Tail Effectiveness A B-matrix error that was introduced in the system is where the tail has lost half its effectiveness and the response of various controller architecture to this scenario is recorded below. Figure (36) present the responses of the PI controller without adaptation. It can be observed from the q plot that the oscillations are of higher magnitude in this kind of failure due to the role tail effectiveness plays in counteracting the moments generated by the wing and the controller is not able to reduce the error arising because of the tail flapping. This same error, modeling using a PD controller, made the closed system unstable. Hence, no plot is shown.

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Figure 36. PI Controller + Reduced Tail effectiveness Figures (37) and (38) present the responses of the system with ABC adaptation for the PI and PD controller architectures respectively. In both the cases the ANN was able to successfully adapt to the failure but could not damp the oscillations in the system. u

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Figure 38. PD Controller + ANN The same error was introduced with the OCM adaptations. Only the controller iterations with learning rates 10000 and 50000 are presented here. From figures (39) and (40), we can observe that by increasing the learning rate we can reduce the oscillations in system response but the tracking error becomes larger.

Figure 39. OCM-10000



Figure 40. OCM-50000 E. Time Delay Error Metric Study In order to identify more suitable learning rates and gains in OCM architecture, a time delay margin study using a tracking error metric was done. The learning rates considered here are same as considered in above sections. The graphs of the learning rates 50000 and 10000 have been terminated at the final value of time delay for which the system was stable, i.e, the error metric value becomes infinity after those points. It can be observed that lower learning rate has a higher time delay than the higher learning rates. Learning rate of 250 , though giving a higher time delay margin could not provide improvement in performance of the PI controller while the learning Figure 41. error metric study rate 50000, though has a lower time delay margin provides a good performance improvement to PI controller in form of damping of system response while also providing a good tracking performance. The learning rate of 10000 lies in-between with a higher time delay than 50000 and acceptable performance. It has to be noted that the for this study the PI controller gains were fixed at the values chosen before, hence in order to increase these time delay margins we can vary those gains to have an acceptable tracking performance combined with higher time delays.

IX.

Conclusion

This paper presented a general model of an ornithopter combining the unsteady aerodynamics with consideration of wing shape and multi-body flight dynamics model. This work further develops a Model Reference Adaptive Controller for pitch rate command using the Adaptive Bias configuration as the adaptive element. This controller was subject to various failures in the form of A matrix error and reduced tail effectiveness. The controller architecture consisted of a Model follower, PI/PD controller, Dynamic Inverse controller that tracks pitch, velocity accelerations 28 American Institute of Aeronautics and Astronautics


and an adaptive element. These, in combination, generated the necessary tail setting angle commands. These commands were the inputs to the nonlinear aircraft simulation of a model similar to “Slow Hawk” ornithopter. Each design involved tuning the PI/PD controller gains along with the learning co-efficient of the ABC controllers,𝜂. In addition, this work also adopted the OCM architecture originally developed for fixed wing aircraft and studied the system response to failures. Subsequently, a time delay error metric study was performed for various versions of this controller in order to identify optimal learning rates. Each of the architectures were presented with a commanded pitch rate doublet. Time responses for pitch rate, angle of attack, pitch angle, forward velocity, tail setting angle were shown for each of the test points. The following are the conclusions for the controller’s performance in flapping wing aircraft. A. PI Controller with Adaptive Bias Correction The PI Controller architecture was able to adapt to the modelling error and tracked the commanded pitch rate doublet in absence of any kind of failures but in presence of failures, the PI controller tracked the command with a steady state error. The adaptive element was able to correct this and successfully tracked the commanded doublet in the presence of modelling error and tail effectiveness error. One of the observations was that the failure modes did not create any instability in the system and the PI controller was able to correct the error. B. PD Controller with Adaptive Bias Correction The PD Controller architecture was able to track the commanded pitch rate doublet in absence of any kind of failures but in the presence of failures, the PD controller was not good at tracking the command and in the special case of reduced tail effectiveness it was not able to stabilize the system. The adaptive element was able to correct this and successfully tracked the commanded doublet in presence of modelling error and tail effectiveness error, it made the system’s response to be similar to a first order system but increasing 𝐶𝑚𝛼 error led to increase in the rise time of the adaptive system. C. OCM Adaptaion The OCM controller architecture was also studied as part of this work and it was observed that this controller requires higher learning rates in order to perform well. The advantage of this controller was seen to be its ability to dampen out the oscillations in system response. It was observed that with very low learning rates the system mimicked the original PI system and with higher learning rates it showed an improved response. With higher learning rates it was observed that the tracking ability of the system was degraded and the time delay that the controller can accept also degraded. Hence, in order to optimize this controller for this aircraft the proportional and integral gains should also be varied so that the entire system with PI and OCM adaptation together gives us the required tracking performance while increasing the time delay. References 1

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Billingsley, D., Slipher, G., Grauer, J., and Hubbard, J. "Testing of a passively morphing ornithopter wing," AIAA Paper Vol. 1828, 2009. 11 Chen, B.-H., Chen, L.-S., Lu, Y., Wang, Z.-J., and Lin, P.-C. "Design of a Butterfly Ornithopter," Journal of Minjiang Science and Technology Vol. 19, No. 1, 2016, pp. 7-16. 12 Brown, R. H. J. "THE FLIGHT OF BIRDS," Biological Reviews Vol. 38, No. 4, 1963, pp. 460-489. doi: 10.1111/j.1469-185X.1963.tb00790.x 13 Chatterjee, S., Templin, R. J., and Campbell, K. E., Jr. "The aerodynamics of Argentavis, the world's largest flying bird from the Miocene of Argentina," Proc Natl Acad Sci U S A Vol. 104, No. 30, 2007, pp. 12398-403. 14 Dial, K. P., Jackson, B. E., and Segre, P. "A fundamental avian wing-stroke provides a new perspective on the evolution of flight," Nature Vol. 451, No. 7181, 2008, pp. 985-989. 15 Hedenström, A. "Aerodynamics, evolution and ecology of avian flight," Trends in Ecology & Evolution Vol. 17, No. 9, 2002, pp. 415-422. 16 Pennycuick, C. J. "Power Requirements for Horizontal Flight in the Pigeon Columba Livia