Flight Dynamics Modeling and System Identification of ...

Flight Dynamics Modeling and System Identification of a Cyclocopter in Forward Flight Elena Shrestha [email protected] Graduate Research Assistant University of Maryland Moble Benedict [email protected] Assistant Professor Texas A&M University

Vikram Hrishikeshavan [email protected] Assistant Research Scientist University of Maryland

Derrick Yeo [email protected] Research Associate University of Maryland

Inderjit Chopra [email protected] Alfred Gessow Director and Distinguished Univ. Professor University of Maryland

Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering, University of Maryland, College Park, MD 20740 ABSTRACT This paper describes the system identification methodology and the identified flight dynamics model of a cyclocopter micro air vehicle (MAV) in forward flight. The cyclocoper utilizes two cycloidal rotors (cyclorotors), a novel horizontal-axis propulsion system with rotating wings. The forward flight control strategy employs independent rpm control of all three rotors and thrust vectoring of two cyclorotors. Unlike a conventional helicopter, a cyclocopter is propelled in forward flight purely by thrust vectoring. This allows the vehicle to maintain a steady, level attitude in forward flight. Even though such a strategy could facilitate power-efficient forward flight, it is accompanied by a strong yaw-roll cross coupling, which is in addition to the inherent gyroscopic coupling that is also observed in hover. To understand these couplings and characterize the bare airframe dynamics, a 6-DOF flight dynamics model of the cyclocopter was extracted using time domain system identification techniques. The existence of a control derivative in the longitudinal translation mode showed that thrust vectoring by phasing the cyclic pitch commands forward speed. After a phasing input, the vehicle achieves steady state within two seconds. The model was able to validate the existence of the inherent roll-yaw coupling in forward flight, which was identified by contributions of roll-yaw coupling stability and control derivatives. The gyroscopic coupling is caused by unbalanced angular momentum and the controls coupling arises from increased propulsive forces at high phase angles. Decoupling methods involve simultaneously mixing roll and yaw inputs in the controller. With onboard feedback stabilization and roll-yaw mixing strategy, the cyclocopter is able to achieve decoupled dynamics in forward flight.

INTRODUCTION

(Fig. 1). These vehicles have demonstrated stable hover and also steady level forward flight. The Seoul National University has also explored the cyclocopter concept, beyond the MAV-scale, by conducting a tethered flight of a cyclocopter UAV weighing 100kg (Refs. 6, 7).

In the last decade, the cyclocopter micro air vehicle (MAV) has received significant attention from the rotorcraft community since its inception in the early 20th century. The cyclocopter is an unconventional configuration that uses cycloidal rotors (cyclorotors) as its main source of propulsion. In 2011, the University of Maryland (UMD) demonstrated the first successful stable hover of a cyclocopter weighing 200g (Ref. 1). Since then, the institution has continued to develop several flight-capable cyclocopters (60–800g) operating at low Reynolds number (15,000–55,000) (Refs. 2–5)

The cyclorotor is a horizontal axis propulsion system where the blades are parallel to the axis and perpendicular to the direction of flight. Since each spanwise blade element of the cyclorotor operates under similar aerodynamic conditions (i.e., flow velocity, angle of incidence, Reynolds number, etc.), the blades can be easily optimized to achieve best aerodynamic efficiency. Compared to a conventional edgewise rotor at the same disk loading (thrust/disk area), an optimized cyclorotor has a higher power loading (thrust/power) in hover (Fig. 4) (Ref. 8).

Presented at the AHS 71st Annual Forum, Virginia Beach, VA, May 5–7, 2015. Copyright © 2015 by the American Helicopter Society International, Inc. All rights reserved. 1

Fig. 1. Flight-capable cyclocopters developed at the University of Maryland. Fig. 3. Blade kinematics. cyclorotors, the thrust vectors are tilted forward and the cyclorotors provide both lift and propulsive thrust. Unlike conventional rotorcraft that pitch forward, the cyclocopter is able to maintain a steady and level attitude in forward flight. In addition, the transition from hover to forward flight does not require any changes in configuration. Overall, thrust vectoring provides a mechanically simple and power efficient method of achieving forward flight. The capability of instantaneously setting the thrust vector to any direction perpendicular to the horizontal axis of rotation also improves the maneuverability and disturbance rejection capability of the cyclocopter. This makes the cyclocopter a robust platform for both indoor application in a highly constraint environment and outdoor operations. Previously, UMD conducted a system identification study to evaluate and compare the disturbance rejection capability of the cyclocopter to other MAV platforms in hover (Ref. 10). The study determined that the cyclocopter had a high longitudinal and lateral resistance to velocity perturbation from external forces. In addition, the cyclocopter dynamics are dominated by a destabilizing gyroscopic coupling between the roll and yaw degrees of freedom. Overall, the flight dynamics model derived from the study revealed many key insights into the vehicle’s maneuverability in hover conditions.

Fig. 2. Cycloidal rotor. The geometric blade pitch angle on the cyclorotor is individually actuated through a four-bar pitching mechanism such that the pitch angle is cyclically changed at each point in the circular trajectory. The blades at both the bottom and top-most point in the trajectory have a positive geometric angle, which results in a positive thrust after a full rotation. The resulting time-varying force can be resolved into vertical (lift) and horizontal (drag) components (Fig. 3). The magnitude and direction of the net thrust vector can be changed by varying the amplitude and phase of the cyclic blade pitch.

In 2014, UMD demonstrated the first forward flight of the cyclocopter using only thrust vectoring as means of achieving forward speed (Ref. 11). The study found that both gyroscopic coupling and controls coupling exist in high speed forward flight. The controls coupling arise when the thrust vector is tilted at a high angle and increasing the magnitude of the thrust vector results in both roll and yaw responses. The wind tunnel experiments showed that both couplings increase with phasing and can be decoupled through a simultaneous roll and yaw inputs. Without a flight dynamics model

Thrust vectoring enables the cyclocopter to achieve a steady, level forward flight. A wind tunnel study conducted by UMD demonstrated that the power required to maintain forward flight decreases with forward speed up until the advance ratio of 1 (Ref. 9). The cyclorotor is able to maintain lift in high speed forward flight without employing any additional lift augmenting surfaces. By varying the cyclic pitch of the 2

Table 1. Weight distribution of the 550 grams twincyclocopter.

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% Total 28.1 5.5 12.7 16.4 23.7 13.6 100%

propeller-motor system with a vertical axis of rotation. The nose rotor provides anti-torque control against the large pitching reaction moment from the cyclorotors spinning in the same direction. The cyclorotors are independently driven by a 75 watts out-runner motor through a 7.5:1 single-stage transmission while the nose rotor is directly driven by a 920kV brushless outrunner motor and is fitted with a 3-bladed GWS 9050 propeller. All three rotors are powered by a single 3S 11.1 volt 850 mAh Lithium-Polymer battery. Overall, the vehicle has a lateral dimension of 0.381 meters (1.25 feet), longitudinal dimension of 0.457 meters (1.5 feet), and a height of 0.305 meters (1 foot) (Fig. 6). The weight distribution is provided in Table 1.

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Fig. 4. Power loading (thrust/power) vs. disk loading (conventional micro-rotors vs. optimized cyclorotor) (Ref. 8)

Cyclorotor Design The cyclorotor design has been optimized by leveraging the results of a comprehensive parametric study (Refs. 12, 13). The cyclorotor parameters (rotor radius, blade airfoil, chord, planform, etc.) were varied in a systematic method to maximize power loading (thrust/power). Based on the study, each blade uses a NACA 0015 airfoil, chord/radius of 0.625, and a span of 6.75 inches. In addition, the cyclorotor has a diameter of 0.152 meters (6 inches) and a chord of 0.051 meters (2 inches).

Fig. 5. Power versus forward speed for constant rotational speed of 1740 rpm (13.88 m/s) for level, steady flight (Ref. 9). in forward flight, a thorough comparison between the hover and forward flight dynamics has not been performed.

Another important aspect of the cyclorotor performance is the structural design of the blades. The blades must be able to withstand dominating centrifugal forces because each gram of blade weight amounts to approximately 0.4N of centrifugal load. The blades also need high bending and torsional stiffness to reduce deformations caused by transverse centrifugal loading. Therefore, the blades on the vehicle use a monolithic design with a foam core wrapped in a singly ply of ±45◦ carbon fiber prepreg. The thickness of the NACA 0015 airfoil provides the bending stiffness and the closed cross-section area and carbon fiber orientation provides torsional stiffness. Overall, each blade weighs 8 grams and the four-bladed cyclorotor produces 200g of thrust at 1400 rpm.

The present work aims to develop a flight dynamics model in forward flight using system identification techniques. The goal is to assess the vehicle’s susceptibility to velocity perturbation and compare it against performance in hover conditions. In addition, this research will also study the gyroscopic and controls cross-couplings in different flight modes. This paper first summarizes vehicle system design and controls strategy in forward flight. The methodology used to formulate the flight dynamics model is then presented in detail. Finally, the system identification results and model comparison between the two flight modes are discussed.

Thrust Vectoring Mechanism

CYCLOCOPTER SYSTEM INTEGRATION The vehicle used for system identification is a twin-cylocopter weighing 550 grams that was previously developed at UMD (Ref. 11). The cyclocopter uses two cyclorotors and a nose

Besides the cyclorotor structural properties, previous experimental parametric studies also optimized the blade pitching amplitude and location of the pitching axis. The cyclorotors 3

Fig. 8. Pitching mechanism on the cyclorotor. picted in Fig. 8. Fig. 6. 550 gram twin-cyclocopter.

For the current setup, thrust vectoring is controlled using two BMS-373MG servos that are connected to the pitching mechanism through a 2:1 transmission. Using this gear arrangement, the offset link is rotated and the pitch phasing could be varied up to 120◦ . Virtual Camber and Incidence A previous study at UMD looked at the physics behind the lift and thrust produced by the cyclorotors in forward flight (Ref. 15). The blades on the cyclorotor experience virtual camber effect because of the curvilinear nature of the flow, which introduces a chordwise variation of velocity direction. Therefore, a symmetric airfoil at 0◦ pitch behaves as a virtually cambered airfoil at a virtual incidence (Fig. 9) in a rectilinear flow. The blades experience a negative virtual camber and incidence at the top of the trajectory (Cl < 0) and a positive camber and incidence at the bottom portion of the trajectory (Cl > 0).

Fig. 7. Four-bar linkage based pitching mechanism.

In forward flight, the advancing side of the blades are at the bottom portion of the trajectory while retreating at the top of the trajectory. The blades on the advancing side experience an increase in the net tangential velocity and a reduction in the retreating side. As a result, the blades produce a small downward lift force while operating in the upper half of the trajectory and large upwards lift force on the lower half, leading to an overall net positive lift force (Fig. 10). Since virtual camber is influenced by the chord/radius ratio of the rotor and virtual incidence is also dependent on the chordwise location of the blade pitching axis, the magnitude of lift depend on these parameters and the direction of rotation.

on the 550-gram twin-cyclocopter use a symmetric pitching cycle with a maximum pitching amplitude of 45◦ . The maximum achievable pitching amplitude is limited by the onset of the blade stall. However, due to unsteady flow mechanisms that delay pitch-rate induced stall, the cyclorotor continues to see an increase in thrust even at high angles of attack (45◦ ) (Ref. 14). Previous flow field studies also showed an existence of distinct leading edge vortex and constructive blade vortex interactions which may be augmenting the lift producing capability of the cyclorotor. The blade pitching mechanism is based on a four-bar linkage system where L1 to L4 are the four linkages (Fig. 7). With this mechanical arrangement, as the rotor rotates, the blades pitch passively along the trajectory. The most critical component in the linkage system is the offset link (L2 ). The rotation of the offset link changes the phasing of the cyclic pitching and thereby changes the direction of the thrust vector. The actual pitching mechanism implemented on the vehicle is de-

While the direction of rotation is not important in hover, it is critical to the lift and thrust producing capability of the cyclorotor in forward flight. In the present vehicle, both the cyclorotors spin counterclockwise, resulting in an unbalanced clockwise torque that must be counteracted by the conventional propeller. To obtain a positive resultant lift force, the cyclorotors are mounted aft of the propeller system such that 4

Fig. 9. Virtual camber and incidence in a curvilinear flow

Fig. 11. Schematic showing hover condition with γabs > 0◦ and Φ=0◦ . is controlled by varying the cyclorotor rotational speed. The direction of the thrust vector is rotated through cyclic pitch phasing. To achieve a stable hover, the phase angle must be rotated such that the resultant thrust vector is purely vertical. For the remainder of the paper, the forward flight phase angle (Φ) of the cyclorotor, distinct from the absolute phase angle of each blade, is defined with respect to the phase in hover (γhover ). Φ = γabs − γhover (1)

Fig. 10. Lift production based on virtual camber and inciA forward flight phase angle of Φ = 0◦ corresponds to the dence in forward flight. hover condition when there is a net lift component and zero the blades are retreating at the top and advancing at the bottom propulsive force. The phasing of cyclic blade pitch is critical of the trajectory. to achieving forward flight and will be discussed in the next section.

CYCLOROTOR KINEMATICS IN FORWARD FLIGHT

Attitude Control Strategy The coordinate system used for the cyclorotor is shown in (Fig. 11). The azimuthal position of the blade (Ψ) is measured counter-clockwise from the horizontal axis of rotation and is equivalent to zero when the blades are rightmost side of the circular trajectory. The blade pitch angle (θ ) is the angle between the blade chord and the line tangent to blade path. The absolute phase angle (γabs ) is defined as the relative azimuthal location of the maximum blade pitch angle with respect to the -Z-axis. When γabs = 0◦ , the maximum blade pitch occurs at the top of the circular trajectory and when γabs = +90◦ , the maximum occurs +90◦ clockwise towards the direction of flow.

Fig. 26(a) shows the definition of pitch, roll, and yaw degrees of freedom for the twin-cyclocopter. Stable hover, transition, and forward flight require a unique combination of independent rotational speed of all three motors and thrust vectoring of the cyclorotors for each mode of flight. A positive roll is produced by increasing the rpm of the left rotor and decreasing the rpm of the right rotor (Fig. 25(c)). Finally, yaw is controlled through differential rotation of the two cyclorotor thrust vectors (Fig. 25(d)). The cyclorotors spin in the same direction along the +Y-axis so there is a net angular momentum that induces gyroscopic coupling between the roll and yaw degrees of freedom. To eliminate the gyroscopic coupling, an onboard control mixing was implemented where the roll and yaw inputs were appropriately combined to give decoupled motion.

The lift and propulsive force components are defined as the net aerodynamic forces produced along the -Z-axis and +Xaxis. For the present cyclorotor, the pitch amplitude is kept constant and the magnitude of the net resultant thrust vector 5

(a) Definition of pitch, roll, and yaw degrees of freedom.

Fig. 12. Cyclorotor coordinates at forward flight where γabs > 90◦ and Φ=90◦ . In forward flight, the combined forward flight phase actuation of cyclic blade pitch is utilized to control the forward velocity of the vehicle. Providing a cyclic pitch phasing for both cyclorotors rotates the resultant thrust vectors by the same phase angle. Essentially, an increase of the combined forward flight phase angle would result in a higher propulsive force contribution from the cyclorotors. As in hover, pitch, roll, and yaw moments in forward flight are controlled through propeller rpm, differential cyclorotor rpm, and differential thrust vectoring, respectively. The pilot controls now include throttle (collective cyclorotor rpm), pitch (propeller rpm), roll (differential cyclorotor rpm), yaw (differential phasing), and forward velocity (collective phasing).

(b) Pitch control.

For the cyclocopter, transitioning from hover to forward flight does not involve any configuration changes. By slowly increasing the phase angle, the vehicle is able to steadily increase forward velocity and transition into forward flight. The rate of transition from hover to forward flight depends entirely on the phasing input. When transitioning back to hover, the cyclocopter is able to abruptly come to a stop even while moving at reasonable forward velocities. The phasing of pitch provides complete control authority of the forward velocity.

(c) Roll control.

At a constant forward velocity and phasing, increasing the rotational speed results in an increase of both lift and thrust and consequently, an increase in power. Increasing phasing of the cyclorotors, while keeping both the forward velocity and rpm constant, results in an increase in thrust and a decrease in lift. Therefore, to increase steady level flight speed from a trimmed condition, both the phasing and rpm is simultaneously adjusted. As the forward velocity increases, the required phasing increases almost linearly while the required rpm decreases (Fig. 14).

(d) Yaw control. Fig. 13. Control strategy for twin-cyclocopter.

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Fig. 14. Controls inputs required to obtain trimmed free flight (lift = weight, thrust = drag).

EXPERIMENTAL SETUP Characterizing the flight dynamics model using system identification techniques requires a series of pilot input excitation experiments and corresponding vehicle measurements. During the experiment, multiple lateral, longitudinal, and yaw doublet inputs were provided in separate flights. The amplitude of the doublets were selected such that the vehicle response has an acceptable signal-to-noise ratio without causing a critical crash-inducing response. Over 60 flight tests were conducted in a 75m2 test environment. The maneuvers were only initiated after the vehicle achieved a steady state of 2 m/s forward speed and steady, level attitude.

Fig. 15. VICON motion capture testing setup.

Fig. 16. ELKA, 1.3 gram processor-sensor board developed at UMD.

Capturing pilot inputs and vehicle responses within the limited test volume was challenging as the inputs had to be given only after achieving a steady cruise. In addition, the inputs were limited to doublets instead of a frequency sweep because of: (1) limited flight testing time in the test space, (2) difficulty for the cyclocopter to maintain steady flight conditions at low frequency inputs, and (3) poor signal-to-noise ratio at the lower frequencies. The rest of this section will provide a description of the experimental methodology and sensor instrumentation.

The onboard instrumentation consists of ELKA, a custom micro embedded processor-sensor board weighing 1.3g (Ref. 16). ELKA houses a STM-32 microprocessor with a 32-bit ARM Cortex M3 core for high-end onboard computation tasks and performs all the data processing, feedback control, and roll-yaw decoupling computations (Fig. 16). The MPU-9150 IMU integrated on the board includes tri-axial gyroscopes, accelerometers, and magnetometers. Wireless communications are serviced by an onboard nRF24L01 chip, a

Data Acquisition Setup The first step towards identifying the flight dynamics model is to systematically collect flight data. Both onboard instrumentation and external sensors are simultaneously used to collect position, velocity, and attitude data. The VICON Motion Capture System consists of eight high speed infrared cameras that track the retro-reflective markers placed on the vehicle (Fig. 15). This enables the motion capture system to triangulate the location of the markers inertially within the threedimensional test volume. These infrared cameras capture the translational and rotational states of the vehicle at a frequency of 100 Hz.

Fig. 17. Data flowchart for feeback integration between ELKA and VICON. 7

Fig. 18. Closed-loop feedback system. the measured states from ELKA were down-sampled in order to synchronize with the VICON data.

low-power 2.4 GHz RF transceiver. The board is connected to a commercial Spectrum receiver which communicates with the pilot’s transmitter through a wireless 2.4GHz radio link. This independent connection provides a degree of redundancy because if ELKA fails, the pilot will still be able to control the vehicle. ELKA has a sensor update rate of 500Hz and is capable of streaming vehicle attitude data and actuator controls data to the base station with a short latency.

Bare Airframe Dynamics Typically, system identification is conducted on an open-loop system to determine the bare airframe characteristics of a vehicle. However, since the cyclocopter is dynamically unstable, the closed-loop feedback system remained implemented with low control gains during the experiments. Furthermore, any decoupling techniques and mixing ratios were eliminated during the experiment in order to capture true vehicle dynamics in forward flight. There are two conventional methods of extracting the bare airframe dynamics from a closed-loop system: (1) having a priori knowledge of control law to deduce the closed loop dynamics matrix or (2) directly measuring actuator inputs to the rotors as a result of both pilot and onboard stabilization. Since the actuator inputs were directly available from onboard measurements, the second approach was taken.

Both VICON and ELKA data are simultaneously streamed to the ground station through a wireless IEEE 802.15.4 data link. At the end of each experiment, a fiducial pitch perturbation was manually provided to the vehicle on the ground in order to synchronize the VICON and ELKA data streams. Both data sets were then allocated a common time stamp in the ground station. The system identification data includes the translational states from VICON and the rotational states from ELKA (Fig. 20). Sampling Rate

Closed-loop Feedback System

The pitch (q), roll (p), and yaw (r) attitude rates are provided by the tri-axial gyroscopes and combined with accelerometer data to produce precise attitude measurements. To account for the drift in gyro measurements, a high pass complementary filter is used with a 4 Hz cut-off frequency for the gyro measurements (Refs. 17, 18). Vibrations from the rotary systems affected accelerometer measurements, which are processed through a low pass complementary filter with a 6 Hz cut-off to reject disturbances from the rotor vibrations occurring at significantly higher frequencies than the body dynamics.

On the cyclocopter, closed-loop feedback is implemented using a proportional-derivative (PD) controller as shown in (Fig. 18). The feedback states are aircraft pitch and roll Euler angles (φ , θ ) and the attitude rates (p, q and r). The inner loop vehicle stabilization includes attitude rate damping and attitude proportional feedback control. An outer loop for translational positioning is performed by a remote pilot. The final control inputs to the vehicle actuators are the individual rotational speeds of the rotors and the two servo inputs for thrust vectoring.

The general rule for selecting the sampling rate is (Ref. 19): fs = 25 fmax

For the first method, the bare airframe dynamics are extracted from the closed-loop model through matrix arithmetic. The closed-loop dynamics (ACL ) are first identified using system identification techniques. With knowledge of the control law, the gain matrix (K), and pilot inputs (u), the bare airframe dynamics (A) are:

(2)

where fmax is the maximum rigid body frequency. For the cyclocopter, the rigid body dynamic modes are between 4– 6 Hz so the optimal sampling frequency should be between 100–150 Hz. As previously mentioned, the onboard processor measures the vehicle states at 500 Hz compared to 100 Hz from the motion capture system. As part of post-processing,

A = ACL + BKu 8

(3)

Model Structure

However, the second method is used in this research since the actuator inputs are directly measured. This approach reduces post-processing time needed for system identification.

For a linearized dynamic model in forward flight, a state space model is considered in the general form: x˙ = Ax + Bu

(4)

The A matrix is composed of the stability derivatives that describe the homogeneous system dynamics and the B matrix contains the control derivatives. The coordinate system for the cyclocopter is depicted in Fig. 20. The state vectors are x = [u v w p q r φ θ ] and the control inputs are u = [δlat δlon δthrottle δ ped δ phase ]. δlat refers to the differential rpm of the two cyclorotors, δlon is the rpm control of the propeller, δ ped is differential thrust vectoring, and δ phase is the phasing of the cyclic pitch. These inputs are nondimensional and represent the duty cycle of the PWM signal transmitted to the actuators. δlat , δlon , and δ ped ∈ [−300, 300], δthrottle ∈ [0, 1500], and δ phase ∈ [0, 60]. For comparison with other existing flight dynamics model of MAV platforms, the inputs have been scaled to u ∈ [−1, 1]. As a result, the control derivatives can be compared across vehicle platforms.

Fig. 19. System Identification using time-domain approach.

The most significant simplifications made for the model is that the cyclorotors are governed by rigid-body dynamics. In addition, the longitudinal and heave modes are independent and decoupled from the rest of the dynamics. Further simplification is that the heave mode in forward flight is identical in hover conditions. Time-Domain Approach The time-domain approach was used for system identification of the cyclocopter dynamics because of restrictions in maneuver time and limited frequency of input excitations. The frequency-domain approach generally requires a large amount of flight data with pilot input excitation across multiple frequency ranges. In addition, the time-domain data can be directly interpreted and applied to the physical system. While the advantages of using frequency domain methods are less computation time and applicability to unstable systems, it can fail to identify biases and linear trends in data.

Fig. 20. Forces and moments on the cyclocopter.

Within the time-domain, there are two main regression techniques: (1) Equation-error method and (2) Output-error method. The Equation-error method assumes that the state variables are measured without any error while the outputerror method assumes that the inputs are measured without error and instead matches output information. One disadvantage of the output-error method is that it involves integrating differential equations for each iteration of the modified NewtonRaphson technique and may lead to divergence in parameter estimation for unstable systems (Ref. 19). The Equation-error method is used in this research because it does not require integration of the state equations and will converge to some parameter estimation.

SYSTEM IDENTIFICATION METHODOLOGY System identification is performed using the NASA Langley System Identification Programs for Aircraft (SIDPAC) (Ref. 19). SIDPAC is a software package containing MATLAB scripts and functions of system identification tools. It is capable of performing both frequency-domain and timedomain analysis of flight data. In addition, it contains additional tools for real-time parameter estimation, model structure determination, input design, and data analysis. The first step in system identification is applying engineering simplifications and prior knowledge of the dynamic system to predict the model structure (Fig. 19).

Before applying the equation-error method, the model structure is identified using stepwise regression. The process 9

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Fig. 21. Time history data for pilot excitation and vehicle response.

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Fig. 22. Model structure for cyclocopter in forward flight. begins with a large pool of potential candidate regressors and their calculated correlation. Through forward selection and backward elimination, each regressor is selected or eliminated at each step until there are no more regressors with significant correlation. Afterwards, the model is refined using the equation-error method, which calculates the potential correlation of each regressor based on a linear lease-squares method.

Table 2. Parameter estimation results. Parameter Value F-ratio Longitudinal Mode Xu -0.021 51 X phase 1.55 125 Mu 1.97 31 Mq 0.009 22 Mlon -83.1 96 Lateral and Yaw Modes Yv -0.036 68 Yp -0.001 115 Yr 0.003 242 Lv 4.29 54 Lp 0.097 53 Lr 0.056 194 Llat 76.4 31 Lrud 114 91 Nv 1.14 109 Np 0.051 32 Nr 0.018 149 Nlat -53.9 34 Nrud 19.5 23

RESULTS This section presents results of the system identification, including model structure determination, parameter estimation, and eigenstates. Pilot Excitation and Vehicle Response During the experiment, all decoupling methods were removed in order to observe the vehicle’s inherent roll-yaw coupled responses. As previously mentioned, the longitudinal and heave modes were assumed to be independent and decoupled. In addition, the heave mode was assumed to be identical to that in hover conditions. These assumptions were validated from the flight test results (Fig. 21). Flight tests were systematically conducted by first providing a pure throttle command and observing the vehicle response. Any forward translation was counteracted by either reducing the phasing of the cyclic blade pitch or varying the propeller rpm. After trimming the cyclocopter, level forward flight is then initiated solely through thrust vectoring of the cyclorotors.

effects of data collinearity can be suppressed by omitting the secondary regressors from the regressor pool. For example, a positive roll input induces a clockwise yaw response, which leads to a counterclockwise yaw actuator input from the feedback system. Therefore, δyaw would be present in the regressor pool even though only a roll input was initially actuated. Collinearity reduces the accuracy of parameter estimation and therefore, the secondary regressors must be manually omitted.

All maneuver inputs were provided after the cyclocopter achieved a steady state of 2 m/s flight at a steady, level attitude (θ , φ , ψ=0). From the time history data, increasing the phasing to Φ = 45◦ with a δ phase step input (time = 20 seconds) also causes a step change in velocity (Fig. 23(b)). During the acceleration phase, the initial response is large and acceleration begins to decrease with time. When the phase input is returned, the flight speed returns to the original forward velocity at the same rate.

Model Structure Determination The linear model structure was first determined using stepwise regression and the corresponding stability and control derivatives were identified in the form provided in Fig. 22. From the pool of regressors, the identified model parameters were selected based on the coefficient of determination (R2 ) and Fratio (Fp ). The R2 values exist within [0,1] and increase with the addition of regressors. If the value changes by only 0.5, then the regressor is deemed statistically insignificant and is

Another observation from flight test results is the existence of a dominating roll-yaw coupling for both lateral and yaw inputs. The coupling also causes a secondary response from the feedback system, which leads to data collinearity. The adverse 11

(a) Angular acceleration in pitch

(b) Longitudinal acceleration

(c) Angular acceleration in roll

(d) Lateral acceleration

(e) Angular acceleration in yaw

Fig. 23. Estimated model output compared to flight data.

12

(b) Roll-yaw response.

(a) Roll right control input.

(d) Yaw-roll response.

(c) Yaw control input.

Fig. 24. Gyroscopic cross-coupling between roll and yaw degrees of freedom. (5)

omitted from the model. All of the identified modes had acceptable R2 values between 0.65–0.86. The Fp values indicate the significance of an individual parameter to the overall model. For a 95% confidence interval, regressors with Fp < 20 were not considered. Overall, the final model parameter estimates are provide in Table. 2. Fig. 23 shows good correlation between model predicted results and flight test data. Limitations in total maneuver flight time was from the restricted motion capture testing volume and length of time it took to reach the steady state. With that in consideration, the estimated model parameters are able to adequately characterize the vehicle dynamics.

Compared to the flight dynamics model derived in hover conditions, the only new model parameter in forward flight is the control derivative, X phase . As expected, X phase is positive because phasing input leads to longitudinal translation. In addition, the strength of the control derivative confirms that forward speed is highly sensitive to phasing input. This was also observed during flight tests. The negative longitudinal translation derivative, Xu indicates positive aerodynamic damping. Therefore, the vehicle is able to stabilize a velocity perturbation in the longitudinal direction. The longitudinal coupling term, Mu is destabilizing. If the vehicle is traversing forward, a velocity perturbation will induce a nose-up pitching moment. The positive pitching moment would further decrease the forward velocity. Likewise, a velocity perturbation from the opposite direction would cause the cyclocopter to pitch-down and increase forward speed. This behavior was also observed in the previous wind tunnel study (Ref. 11).

Longitudinal Mode The longitudinal dynamics are identified as: 

   u˙ Xu 0 −g  q˙  =  Mu Mq 0   0 1 0 θ˙   0 X phase  δlon 0  +  Mlon δ phase 0 0

 u q  θ 

Compared to the other longitudinal stability derivatives, Mq is almost insignificant in magnitude and Fp and can be removed from the model. Therefore, the aerodynamic pitch 13

(b) Roll-yaw response.

(a) Roll right control input.

(d) Yaw-roll response.

(c) Yaw control input.

Fig. 25. Controls cross-coupling between roll and yaw degrees of freedom occurring at high phase angles. damping is non-existent and the vehicle requires feedback damping to stabilize the pitch attitude rate. Finally, Mlon is negative because a positive pitch input results in a negative pitch-down moment.

yaw responses. In particular, perturbation to the right induces a clockwise yaw and a roll-right moment, which would further increase the velocity. The yaw response is due to drag acting on the cyclorotor, which is located forward of the c.g. Based on Yp , the cyclocopter is able to stabilize the lateral translation for a given perturbation in the roll rate. A roll-left moment effects a lateral translation to the right, counteracting the perturbation. The opposite is true for a yaw rate perturbation (Yr ). A clockwise yaw results in a translation to the right, further destabilizing the vehicle. In addition, both L p and Nr are unstable.

Coupled Lateral and Yaw Modes The lateral and yaw dynamics are identified as: 

  v˙ Yv Yp  p˙   Lv L p     r˙  =  Nv N p 0 0 φ˙  0 0  Llat Lrud +  Nlat Nrud 0 0

Yr Lr Nr 0     

 g v  p 0   0  r 1 φ δlat δyaw

   

The most significant observation from the model structure is the contributions of Lr and Lrud to the lateral state and N p and Nlat to the yaw state. This is the dominating crosscoupling effect unique to the cyclocopter. The roll-yaw couplings and their respective decoupling methods are described in details in the following subsections.



(6) Gyroscopic Coupling As seen from the model and flight tests, the cyclocopter dynamics are dominated by gyroscopic roll-yaw cross-coupling. This coupling is caused by the unbalanced angular momentum from the cyclorotors spinning in

Similar to the longitudinal translation derivative, the lateral translation derivative, Yv is stabilizing. However, the lateral and yaw coupling terms, Lv and Nv are destabilizing. In forward flight, a velocity perturbation would induce both roll and 14

Yaw input (µs)

0

8

9

10

11

12 Roll Yaw

50 0 −50 7

8

9

10

11

8

9

10

11

20

22

12

24

26

28 Roll Yaw

50 0 −50 20

0

7

−100

12

20

−20 6

0

Roll angle φ (deg)

6

Roll angle φ (deg)

7

100

Rate (deg/s)

−100 6

Rate (deg/s)

Roll input (µs)

100

22

24

26

28

20 10 0 −10 −20 19

20

21

22

23

24

25

26

27

28

29

Time (s)

Time (s)

(a) Decoupled roll response

(b) Decouple yaw response

Fig. 26. Decoupled responses after implementing mixing strategies between roll and yaw degrees of freedom (Ref. 11). the same direction. For a positive roll input, the vehicle responds in both roll right and counterclockwise yaw (Fig. 24). Likewise, for a positive yaw input, the vehicle responds in clockwise yaw and also rolls to the right.

Table 3. Eigenvalue Comparison Mode

In the present vehicle, the cross-coupling is decoupled by mixing roll and yaw inputs in the controller. To achieve a pure roll right response, a roll right input is combined with a clockwise yaw input to cancel out the inherent cross-coupling response. Likewise, for clockwise yaw, the input is appropriately combined with a roll left input. The mixing ratio is only applied to the open-loop pilot input and directly implemented on the onboard controller.

Longitudinal Translation Longitudinal Rotation Lateral-Yaw Translation Lateral-Yaw Rotation

The gyroscopic mixing terms were experimentally found by suspending the cyclocopter about c.g. on a 3-DOF gimbal stand unrestrained in pitch, roll, and yaw. Starting with zero mixing, the constants for both roll and yaw inputs were slowly increased until the vehicle achieved a decoupled response. Experimental results showed a stronger coupling for a roll input, which results in higher mixing ratios. One reason for a stronger roll-yaw coupling could be that the moment of inertia about the x-axis is greater than that about the z-axis.

The mixing ratios kφlat and kφ ped were identified in the wind tunnel. In the previous experimental study, the mixing ratios were iterated until the vehicle dynamics were decoupled at each phasing input. (Ref. 11). After implementing the decoupling strategy, the vehicle is able to achieve decoupled response in forward flight (Fig. 26). Stability analysis in forward flight The eigenvalues of the system provide a better understanding of the vehicle flight dynamics and have been summarized in Table. 3. The eigenvalues in hover were computed in a previous study using the same system identification techniques (Ref. 10). It is interesting to note that the eigenvalues for the longitudinal modes are similar between the hover and forward flight cases, but greatly differ in the lateral-yaw modes.

Controls Coupling The controls coupling at high phase angles is summarized in Fig. 25. The phenomenon can be physically explained by considering the magnitude of vertical (lift) and horizontal (thrust) forces generated by the cyclorotor at high phase angles. As the phase angle increases, the thrust vector is tilted towards the +X-axis and thus thrust increases and lift decreases. A differential change in rotational speed would have a greater effect on yaw than in roll. Likewise, thrust vectoring would effectively cause a roll response.

The cyclocopter is marginally stable in the lateral-yaw rotation mode in forward flight, but shows highly oscillatory and unstable tendencies in hover. This is possibly the result of the roll-yaw cross-couplings previously mentioned. In addition, the presence of the two lateral-yaw translation eigenvalues indicate unstable coupling in the lateral translation and direc-

The mixing scheme for controls coupling is as follows: δφ = [kφlat δlat + kφ ped δ ped ]sinΦ

(7)

δψ = [kψlat δlat + kψ ped δ ped ]sinΦ

(8)

Eigenvalues Hover Forward Flight -3.83 -2.69 1.66±3.05i 1.33±2.32i -2.29 0.133 1.48 1.01 0.28±8.02i -0.027±0.046i

15

general, the longitudinal mode is decoupled and independent of the rest of the modes.

2.5 2

Imaginary Axis

1.5

Longitudinal Translation Longitudinal Rotation Lateral-Yaw Translation Lateral-Yaw Rotation

3. The lateral translation derivative, Yv is also stabilizing. However, the lateral and yaw coupling terms, Lv and Nv are unstable. In forward flight, a velocity perturbation would induce both roll and yaw responses. The yaw response is due to drag acting on the cyclorotor, which is located forward of the c.g.

1 0.5 0 -0.5 -1

4. Based on Yp , the cyclocopter is able to stabilize the lateral translation for a given perturbation in the roll rate. The opposite is true for a yaw rate perturbation (Yr ). A clockwise yaw results in a translation to the right, which causes the vehicle to diverge from the equilibrium state.

-1.5 -2 -2.5 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

5. The model was able to validate the existence of the inherent roll-yaw coupling in forward flight, which was identified by contributions of Lr and Lrud to the lateral state and N p and Nlat to the yaw state. These are the result of combined gyroscopic coupling and controls coupling at high phase angles. Interestingly,

Fig. 27. Pole location for the cyclocopter in forward flight. tional modes from trimmed flight. Based on the locations of the system poles, the overall bare airframe dynamics of the cyclocopter is inherently unstable (Fig. 27). Therefore, the cyclocopter requires feedback regulation to achieve trimmed flight.

6. The gyroscopic coupling is caused by the unbalanced angular momentum from the cyclorotors spinning in the same direction. The controls coupling is due to the increase of magnitude of propulsive forces at high phase angles. As a result, thrust vectoring has a larger impact on the roll moments and subsequently, changes in rotor RPM have a greater influence on yaw moments.

CONCLUSIONS The objective of this research was to develop a flight dynamics model in forward flight using system identification techniques. The 6-DOF model would allow assessment of the vehicle’s susceptibility to velocity perturbation. This would enable future comparative study of disturbance rejection capability across multiple MAV platforms. The flight dynamics model was extracted using the time-domain approach based on measurements from doublet inputs across various modes. The model structure was initially identified using stepwise regression and the parameters were estimated using the equation-error method. All maneuver inputs were provided after the cyclocopter achieved a steady state of 2 m/s flight at a steady, level attitude (θ , φ , ψ=0). The following is the summary and some of the specific conclusions drawn from this study:

7. Decoupling methods involve simultaneously mixing roll and yaw inputs in the controller to achieve a pure response. Experiments showed a stronger coupling for a roll input, which results in higher mixing ratios. One reason for a stronger roll-yaw coupling is that the moment of inertia about the x-axis is greater than that about the z-axis. ACKNOWLEDGEMENT This research was supported by the Army’s MAST CTA Center for Microsystem Mechanics with Dr. Brett Piekarski (ARL) and Mr. Chris Kroninger (ARL-VTD) as Technical Monitors. Authors would like to thank Dr. Greg Gremillion and Hector Escobar for their help in setting up the system identification experiments. Authors would also like to thank David Hairumian for his help in vehicle construction.

1. The existence of the control derivative, X phase shows that phasing input commands forward speed. From flight tests, it was observed that increasing the phasing with a δ phase step input directly results in a step change in velocity. During the acceleration phase, the initial response is large and acceleration begins to decrease with time until the vehicle reaches steady state after two seconds. When the phase input is returned, the flight speed returns to the original forward velocity at the same rate.

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2. The negative longitudinal translation derivative, Xu indicates positive aerodynamic damping. Therefore, the vehicle is able to stabilize a velocity perturbation in the longitudinal direction. However, the aerodynamic pitch damping (Mq ) is non-existent and the vehicle requires feedback damping to stabilize the pitch attitude rate. In

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