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Abstract

The paper discusses flight-test-based system identification of a compact, re-configurable, rotary-wing micro air vehicle capable of sustained hover and could potentially be launched from a 40-mm grenade launcher. The objective was to extract a linear time-invariant model of the system to gain an understanding of a novel coaxial helicopter. The vehicle design features a cylindrical fuselage, coaxial rotors with foldable/deployable blades, thrust-vectoring mechanism for pitch/roll control, and differential rotational speed for yaw control. Flight experiments were conducted to excite the vehicle’s longitudinal, lateral, directional, and heave modes from a hovering state. A linearized state-space model was extracted from the flight test data. The model showed that the lateral and longitudinal dynamic modes were decoupled from each other and from the other modes. Due to the axisymmetric vehicle design, the longitudinal and lateral stability and control coefficients and their eigenvalues were nearly identical. All the aerodynamic damping terms were negative and stabilizing except for the pitch and roll acceleration modes, which necessitated the need for pitch and roll feedback control.

Keywords

System identification flight testing coaxial rotor micro air vehicle tube-launched thrust-vectoring

Introduction

Micro Air Vehicles (MAVs) find their use in a variety of applications such as intelligence, surveillance, reconnaissance (ISR), as well as search and rescue missions. MAVs provide robust platforms for real-time video or other sensor feeds to the ground operator. The size and maneuverability of rotary-wing MAVs make them an especially promising solution for ISR in crowded urban environments. Despite these advantages, the altitude, range, and endurance of these predominantly electric platforms are restricted by the low energy density of current battery technology.^1,2 For example, the energy density of commercial lithium-ion cells can reach 200 Wh/kg in ideal settings; however, this energy density can decrease to 150 Wh/kg at the pack level in real conditions.^3–5 Limited battery capacity places more stringent constraints on rotary-wing systems compared to fixed-wing aircraft due to the higher power requirement for hovering. Often, rotary-wing MAVs have a hover endurance of less than 20 minutes,^1,2,6 making it imperative to consider unconventional approaches to improve the effectiveness of hover-capable platforms for ISR missions.

Although directly improving the endurance of MAVs poses a significant challenge, the overall range could be increased by launching the vehicle as a projectile toward the desired target location. Instead of draining a significant portion of the onboard battery energy to reach the intended altitude and range, the projectile phase of flight would allow for saving the stored energy until the vehicle reaches its designated operating location. The projectile phase also facilitates the rapid deployment of the vehicle since the flight time between the launch and target location is reduced because the deployment speed is limited by the launch speed rather than the maximum vehicle forward flight velocity. From there, the deployment of rotors allows the vehicle to transition to hover mode and perform its mission. This mission profile is presented in Figure 1.

Figure 1.

Gun-launched MAV concept: Flight phases.

The proposed projectile launch technique can be decomposed into several distinct phases of flight. The vehicle begins in a folded state and is inserted into the launcher. Once launched, the vehicle enters the projectile or ballistic phase, and soon after, a protective cap is released from the vehicle. The cap is intended to protect the vehicle from the explosive charge used for the launch. After the vehicle reaches the apex of the flight path, the rotor system is deployed, and the rotors start spinning. Next, the crucial transition from the passive projectile phase to controlled powered descent begins with the application of rotor braking thrust as the vehicle reaches the target altitude and range. Once sufficiently slowed, hovering flight is attained, and the vehicle can perform its desired mission and fly back to the operator if necessary.

Previous Tube-launched concepts

There are a few existing military platforms that take advantage of a projectile flight phase for rapid deployment and ease of launch without runways; however, most of these are fixed-wing platforms such as Raytheon’s Coyote⁷ and AeroVironment’s Switchblade.⁸ Although fixed-wing platforms tend to have higher endurance compared to rotary-wing ones, fixed-wing vehicles lack the ability to focus on a single target for an extended duration or navigate in confined spaces. Additionally, almost all the existing platforms utilize specialized launching systems, which can be cumbersome for ground personnel to transport. This equipment must be added to the 87–127 lbs of gear carried by the typical warfighter.⁹ Leveraging existing equipment, such as a 40-mm grenade launcher, would significantly increase the convenience and practical applicability of air-launched systems.

In recent years, the development of hover-capable, tube-launched platforms has begun to emerge, such as the Streamlined Quick Unfolding Investigation Drone (SQUID)^10,11 developed by Caltech and the Gun-launched MAV (GLMAV)^12,13 developed by the French-German Research Institute at Saint Louis, and the Tube-Launched MAV (TLMAV) developed at Texas A&M University.¹⁴ A size comparison of these platforms can be seen in Figure 2. The SQUID is a ballistically launched quad-copter with an 83 mm (3in) folded diameter that can be fired from a moving platform. However, the GLMAV concept is a significantly larger coaxial rotary-wing design with a folded diameter of 80 mm (3in). However, the TLMAV uses a mechanically simple thrust-vectoring mechanism instead of a swashplate. Moreover, the TLMAV is significantly smaller (folded diameter of 52 mm) than the other three concepts. While both coaxial-rotor and multi-rotor-based configurations could be scaled down to fit within a grenade launcher, a study by Wereley and Pines¹⁵ determined that a coaxial rotor system was a better candidate for the proposed mission profile. This assessment was based on factors such as efficiency, folding compactness, and ease of packing.

Figure 2.

Size comparison of current tube-launch systems.

Technical barriers

The overarching goal of this research effort is to develop a coaxial rotor-based MAV that could be launched from a grenade launcher. However, developing this platform presents some key technical barriers that stem from scaling the vehicle down to fit within the barrel of a grenade launcher that has an inner diameter of 40 mm, in-flight reconfiguration (passive rotor deployment from a folded state) and the complex dynamics experienced during the transition from projectile to hovering mode. Some specific challenges include (1) designing and building a compact rotorcraft with an outer diameter of less than 40 mm, (2) foldable coaxial rotor blades with passive unfolding strategies that avoid rotor collision, (3) simplified and compact swashplateless pitch, roll and yaw control strategies, (4) ultralight-weight autopilot with a small footprint, (5) ability to handle the high accelerations of an explosive take-off, (6) passive and active control strategies ensuring stable attitude dynamics in flight, especially during the transition from the projectile mode to the hovering helicopter mode, (7) optimizing the rotor-motor-ESC (electronic speed controller) combination to maximize hover endurance, and (8) understanding and improving the controllability and disturbance rejection (gust tolerance) of the vehicle in hover for improved robustness in adverse conditions.

The platform’s compactness, in-flight reconfiguration, and transition requirements necessitate a multifaceted approach to the vehicle design. The size constraint was slightly relaxed to facilitate development, and an intermediate prototype was developed. The intermediate prototype had an outer diameter of 52 mm instead of the required 40 mm diameter. Future iterations will miniaturize this intermediate prototype by optimizing and scaling the components for the final 40-mm design. We have previously demonstrated the ability of the 52-mm TLMAV prototype to achieve a stable projectile phase, deploy the rotors, and then successfully transition from the projectile phase to a stable hover following a vertical launch.¹⁴ An experimental parametric study was performed to examine the performance of the coaxial rotor at low Reynolds numbers.¹⁶ The objective of the present paper is to understand the flight dynamics of the TLMAV using a linear time-invariant (LTI) model extracted via flight testing based system identification.

Figure 3.

Coaxial helicopter MAVs. (a) Tigermoth UAV²⁷; (b) Micro coaxial helicopter testbed³¹; (c) COTS Coaxial helicopter³²; (d) muFly³³

Figure 4.

Tube-launched MAV in hovering flight.

Previous MAV system identification

The creation of linear time-invariant (LTI) models for MAV-scale vehicles is an important step in understanding and predicting their flight dynamics before pursuing more complex nonlinear modeling. These models provide initial insight into system stability, controllability, and dominant dynamic modes, forming a foundation for subsequent control design. Additionally, they offer a simplified framework for validating experimental data and benchmarking control strategies before applying them to nonlinear or adaptive systems.

System identification can be performed in the time domain or frequency domain to create a mathematical model of a system. Time domain system identification takes the time history of the inputs and outputs of a system to extract a system model. The input excitations are generally in the form of step inputs and impulse responses. This approach is particularly well-suited for systems with limited test space or those prone to instability, as it allows for controlled, transient input sequences and avoids the need for sustained excitation. System IDentification Programs for AirCraft (SIDPAC)¹⁷ is often used to perform time domain system identification, where the time history of inputs and outputs is correlated using stochastic estimation methods. Alternatively, frequency domain requires control inputs across a range of frequencies to generate spectrally rich data to extract a model using Fourier or spectral analysis. Comprehensive Identification from FrEquency Responses (CIFERR®)¹⁸ is often used to generate automated inputs including chirps and doublets across a range of frequencies and analyze the vehicle response in the frequency domain. Additionally, these inputs can cause an unstable vehicle platform to crash if adequate space is not available for the vehicle to settle after an excitation. Due to limited test space, the risk of instability, and the complexity of automating frequency sweeps, time domain system identification was the more practical choice for the TLMAV.

The linear model can be used as an initial condition for developing a non-linear model. These models could be open-loop or closed-loop flight dynamics models. Open-loop models describe the system’s dynamics without considering feedback or control inputs, making them useful for analyzing the bare airframe response under specific conditions, as well as in identifying the stable and unstable modes of a system. In contrast, closed-loop models incorporate feedback control to represent how the aircraft, along with the control system, responds to various inputs and disturbances during flight. The open-loop and closed-loop models provide a comprehensive understanding of aircraft stability, controllability, and response to external factors, aiding in the design and optimization of flight control systems for better performance.

Before a closed-loop model can be derived, an open-loop model is often identified. The identification of the open-loop dynamics via flight testing tends to be challenging for MAV-scale rotorcraft because many of these systems are open-loop unstable.^19–25 These vehicles could be difficult for a pilot to fly and control without the assistance of a stability augmentation system. Therefore, system identification of unstable systems gravitates towards flight experiments performed on reduced degree-of-freedom test stands or experiments performed with reduced flight controller gains in free flight.

System identification of NASA’s Mars Helicopter,²⁶ the Tiger Moth,²⁷ and the GLMAV²⁸ were performed using a combination of test stand and flight test experiments. The identification of MAV-scale cyclorotors and flapping wing vehicles was performed using flight test data with reduced controller gains.^21,22,29,30

There have been previous studies on the system identification of MAV-scale coaxial helicopters.^31–35 Most of these platforms used a flybar on the upper rotor for passive roll/pitch stability and only had cyclic control on the lower rotor for pitch and roll control. However, the current coaxial-rotor-based TLMAV uses direct thrust-vectoring for pitch and roll control, which has not been systematically studied in the past. Therefore, developing a flight dynamics model of the TLMAV in hover is the focus of the present paper, and this will lead to a better understanding of this unique platform. This model will have a broader impact beyond the present TLMAV platform because it is relevant (at least qualitatively) to any coaxial rotorcraft using thrust-vectoring based control.

Vehicle overview

The previous work by the authors details the design, development, and flight testing of the current TLMAV prototype, which is a highly compact coaxial rotorcraft MAV amenable to tube-launching and utilizes a thrust-vectoring mechanism for pitch and roll control.^14,36 The vehicle successfully demonstrated a vertical launch from a pneumatic cannon, followed by a stable projectile phase utilizing the fins, passive rotor unfolding, and a final transition to a stable hover.

The vehicle designed for this study was modified from the previous configuration to include a set of Vicon markers for the system identification experiments. The layout of the experimental platform and the weight breakdown are provided in Figure 5 and Table 1, respectively. The vehicle configuration includes the thrust-vectoring mechanism, propulsion system, avionics, battery, and fuselage assembly. The thrust-vectoring mechanism enables pitch and roll control using a two-axis gimbal actuated by two independent servos. Control moments are created by offsetting the thrust axis from the center of gravity (CG). The propulsion system consists of the coaxial motor unit, electronic speed controllers (ESCs), and counter-rotating, folding rotors. The rotors and coaxial motor unit produce thrust and yaw control, using collective and differential rotational speed control, respectively. The avionics system is comprised of a 1.7-gram custom-designed autopilot, receiver, and telemetry module. The autopilot obtains pilot commands from the receivers, which are relayed to the feedback controller. Actuator and motor commands are also sent by the autopilot, and flight telemetry is sent via the wireless communication module to a ground station. All of the vehicle subsystems are powered by a 3-cell, lithium polymer battery housed in the lower section of the vehicle.

Figure 5.

Tube-launched MAV used for system identification experiments.

Table 1.

Weight breakdown of the vertical launch vehicle.

Components	Mass (g)	% Total
Propeller and Hubs	22	6.4
Motor	62	18.0
Fuselage	79	22.9
Battery	102	29.6
Electronics	34	9.9
Servo Actuators	25	7.2
Two Axis Gimbal	9	2.6
Vicon markers and attachments	12	3.5
Total	345 g	100.0 %

Thrust-vectoring mechanism

The thrust-vectoring mechanism is a mechanically simpler way to implement pitch and roll control on a coaxial helicopter platform. The mechanism consists of a dual-axis gimbal, which provides the ability to tilt the thrust vector in two orthogonal directions using two independent actuators. A cutaway of the vehicle depicted in Figure 6 reveals the gimbal, control rods, and actuators. The same figure also shows an example of a pitch moment being produced by titling the thrust vector away from the center of gravity (CG).

Figure 6.

Cutaway showing the two-axis gimbal and the reaction moment produced about the CG

The combination of the dual-axis gimbal with the servo actuators allows the thrust vector, which normally passes through the CG, to be pointed forward/aft (pitch control) or left/right (roll control), similar to a teetering rotor. This tilt of the thrust vector creates moments about the CG that cause the vehicle to rotate and translate in the desired direction. An example of a pitch moment being produced by the thrust-vectoring mechanism is shown alongside the cutaway in Figure 6. The maximum tilt angle of 30 degrees in either pitch or roll is limited to prevent the lower rotor from impacting the body.

Propulsion

The propulsion system for the vehicle consists of a specialized counter rotating-motor unit, electronic speed controllers (ESCs), and folding rotors. The motor unit is fastened to the inside of the inner gimbal ring. The combined thrust of the two rotors provides enough excess thrust for flight and control.

The coaxial motor unit further reduced the mechanical complexity of the current TLMAV configuration. The motor unit consists of two brushless DC motors with windings mounted to a common body, as shown in Figure 7. The lower motor drives the upper rotor, while conversely, the upper motor drives the lower rotor. A steel shaft connects the lower motor to the upper rotor while bypassing the upper motor. A set of bearings isolates the connecting shaft from the upper motor. On the other hand, the upper motor directly drives the lower rotor, which is bolted to the upper motor casing. Mechanically decoupling the two motors in this manner allows the propellers to rotate independently and provides separate interfaces for the ESCs. Additionally, the independent control of the motors allows for yaw control through differential rpm of the fixed-pitch rotors.

Figure 7.

Diagram of the counter-rotating motor unit.

Another key component of the vehicle design is the pair of counter-rotating folding rotors, which were designed to conform to the body of the vehicle when folded. Since the rotor blades needed to fold against the body, a circular-cambered airfoil was chosen to reduce the volume needed by the slots for the blades. This also allowed the blades to be concentric with the body when folded. Moreover, thin, circular camber airfoils have been shown to be more efficient for MAV scale rotors operating at low Reynolds numbers.^37,38 Additional information about the propulsion system can be found in [Ref.¹⁴].

Avionics

The avionics system consists of an autopilot, telemetry module, radio receiver, and battery. All of these components are responsible for receiving commands from the pilot and translating the input to control outputs sent to the actuators, as well as transmitting flight test data to a ground station. The autopilot is the crucial link that connects all of these components together. The vehicle’s systems are powered by a 3-cell lithium polymer battery.

An autopilot called ELKA-R was selected for its small footprint (one-inch square) and the ability to be customized for various vehicle platforms.³⁹ A size comparison of the 1.7 g board is shown next to a quarter in Figure 8. The ELKA-R is equipped with an STM32F405 microprocessor for all onboard computations tasks. The integrated MPU9250 is equipped with tri-axial accelerometers, gyroscopes, and magnetometers. These sensor measurements are used to determine the attitude of the vehicle. The board supports up to 12 Pulse Width Modulation (PWM) actuators when two of the three Universal Asynchronous Receiver/Transmitter (UART) channels are converted to four PWM channels. The PWM channels are used to send control signals to the motors and actuators. The three UART channels can be used to read sensors or send data to other devices, such as the telemetry module and receiver.

Figure 8.

Compact 1.7g ELKA-R autopilot.³⁹

The telemetry module is an XBee 3 Pro Zigbee 3.0 that operates at 2.4 GHz and is connected to the autopilot using a breakout board. This module allows ELKA-R to transmit data such as attitude, gyroscope measurements, and the actuator signals to a second XBee connected at the ground station. The module also receives information from the ground station, such as updated controller parameters and actuator trim values.

The pilot’s inputs are routed from the transmitter to the receiver, which is directly connected to the autopilot. A Taranis X9D transmitter is used to relay pilot commands, and it is capable of sending up to nine different control inputs at the same time. These inputs are encoded and sent to a FrSky R-XSR PPM receiver capable of reading eight different input signals to be sent to the autopilot. The autopilot processes the pilot commands and sends signals to the appropriate actuators.

Control methodology

For system identification experiments, ensuring that the vehicle could be flown in a stable and controlled manner was imperative. From previous flight testing, it was noted that the vehicle appeared unstable in open-loop in both pitch and roll. Attitude stabilization of the vehicle was accomplished using a cascaded proportional-integral-derivative (PID) controller.

Pitch, roll, and yaw control is accomplished by a combination of rotor plane tilting and rotor rotational speed control. As previously mentioned, tilting the thrust-vector generates pitch and roll moment about the CG, as shown in Figure 6. Yaw control is accomplished by commanding differential rotational speeds between the individual motors of the counter-rotating motor unit. The reaction torque produced about the CG by the differential aerodynamic rotor torque generates a yaw rate as depicted in Figure 9. In trimmed hovering flight, there is no gyroscopic coupling; therefore, the pitch and roll dynamics are decoupled from the yaw dynamics.

Figure 9.

Yawing moment created by differentially varying the rpm of the motors.

The pitch and roll attitudes of the vehicle were stabilized using a cascaded proportional–integral–derivative (PID) controller. The gyroscopes on the autopilot measure the body axis rates from which the pitch and roll attitudes are estimated via integrating the gyroscope measurements while using a complementary filter that uses the accelerometer data as a stable bias to reduce drift.^40,41 The control structure is shown in Figure 10(a). Starting from the outer loop for pitch and roll control, the pilot commands a desired pitch and roll attitude using an RC transmitter. The pitch and roll sticks on the transmitter are mapped to produce the desired outputs between ±30 degrees. From there, the error between the desired and estimated attitude is scaled by the outer-loop proportional gain to get the angular desired rate. The desired rate is then passed as the setpoint for the inner loop. The error between the desired and measured rates is passed through a PID feedback controller in the inner loop. The controller output is directly converted to servo commands for the thrust-vectoring mechanism that initiates the desired correction to the vehicle.

Figure 10.

Controller feedback diagram. (a) Pitch and roll attitude feedback; (b) Yaw rate feedback.

The yaw rate of the vehicle was stabilized by just a PID controller. The control structure is shown in Figure 10(b). Since a heading estimator has not been implemented on the vehicle, the pilot commands a desired yaw rate measured from the gyroscopes instead. The yaw stick on the transmitter is mapped between ± 250 degrees per second. As before, the desired rate is compared against the measured yaw rate, and the difference is passed on to a PID feedback controller. The controller’s output is a differential yaw command that increases rpm of one rotor while reducing rpm of the other. The torque imbalance of the rotors produces a yaw angular acceleration. However, for the system identification experiments, the desired yaw rate was set to zero, and the pilot command was linearly added to the final control output so that the pilot yaw command corresponded directly to a yaw disturbance. Since the inertia about the vehicle’s vertical axis is significantly smaller than the pitch and roll moments of inertia, vibrations generated from rotors tended to affect the yaw feedback adversely. Therefore, the gyros measurements needed to be filtered below the rotor operating frequencies, and the derivative term of the controller was set to zero to mitigate excessive yaw oscillations.

System identification methodology

System identification was performed on the vehicle in hovering flight with the goal of creating a linearized flight dynamics model of the open-loop response for the same flight condition. The model will be used to get a better understanding of the design and control of a coaxial helicopter platform that uses a thrust-vectoring mechanism for pitch and roll control. The insights gained should help guide the future development of these types of aircraft. Once a system model is derived, robust and adaptive controllers could be developed to augment the stability and controllability of the system to reject disturbances.

Model structure

The linearized flight dynamics model developed for the vehicle was formulated in the continuous time-invariant state-space form:

$\begin{matrix} \dot{x} \dot{x} = A x x + B u u \\ y y = C x x + D u u \end{matrix}$ (1)

The state vector $x x = [u, v, w, p, q, r, ϕ, θ, ψ]^{T}$ includes the body-axis velocities ( $u, v, w$ ), body-axis angular rates ( $p, q, r$ ), and the attitude Euler angles ( $ϕ, θ, ψ$ ). The time derivative of the state vector is represented by $\dot{x} \dot{x} = [\dot{u}, \dot{v}, \dot{w}, \dot{p}, \dot{q}, \dot{r}, \dot{ϕ}, \dot{θ}, \dot{ψ}]^{T}$ . The control input vector $u u = [δ_{l a t}, δ_{l o n g}, δ_{r u d}, δ_{t h r}]^{T}$ contains the lateral (roll), longitudinal (pitch), directional (yaw), and heave (thrust) control inputs. Each of the control inputs corresponds to the final pulse width modulation (PWM) signal sent to the actuators and is scaled such that $- 1 \geq δ_{n} \geq 1$ where n corresponds to an independent actuator input. The sign convention for control inputs was selected such that positive inputs produce a negative angular rate, but the throttle input was defined as positive for a negative heave velocity. The A matrix is the state transition matrix that describes the vehicle’s open-loop dynamics. Each term in the matrix corresponds to the aerodynamic coefficients. The B matrix corresponds to the control coefficients that describe how the control input vector affects the system dynamics. The output vector is $y y$ , and based on the measured states, the output vector is equal to the state vector in this case. The output matrix, C, maps the state vector to the output vector, and D is the feedback or feedthrough matrix.

For a lower-order linear analysis, it is assumed that the time derivative of each of the states is a linear combination of the states and control inputs. In this form, the coefficients of the A and B matrices can be determined from the solution to the least-squares problem. This study was limited to a time-domain analysis due to the unstable nature of the system and the limited space for flight testing. This prevented the collection of spectrally rich flight test data needed for frequency domain system identification. The flight test data was processed using a software package called SIDPAC, which was developed by Morelli for time domain system identification.¹⁷ The theory behind the software is discussed in detail in Morelli’s companion book for the software package.⁴² This software has been used to identify models for fixed-wing aircraft, rotorcraft, and flapping wing vehicles.^21,22,29,30

A model for the system was postulated for hovering flight based on previous flight tests of the vehicle. It was observed that the longitudinal and lateral dynamic modes were decoupled from each other. Also, the longitudinal and lateral dynamics were decoupled from the directional modes. Both pitch and roll control inputs result in translations and rotations about their respective axes. A yaw input results in a rotation about the yaw axis, and a heave input results in only translation in the z-direction. The gravity terms, denoted by g, were populated in the $\dot{u}$ and $\dot{v}$ equations. These observations were later tested during the development of the final LTI model. With this prior knowledge, a postulated model is given as follows:

$\begin{matrix} (\begin{matrix} \dot{u} \\ \dot{v} \\ \dot{w} \\ \dot{p} \\ \dot{q} \\ \dot{r} \\ \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}) = [\begin{matrix} X_{u} & 0 & 0 & 0 & X_{q} & 0 & 0 & - g & 0 \\ 0 & Y_{v} & 0 & Y_{p} & 0 & 0 & g & 0 & 0 \\ 0 & 0 & Z_{w} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & L_{v} & 0 & L_{p} & 0 & 0 & 0 & 0 & 0 \\ M_{u} & 0 & 0 & 0 & M_{q} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & N_{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{matrix}] (\begin{matrix} u \\ v \\ w \\ p \\ q \\ r \\ ϕ \\ θ \\ ψ \end{matrix}) \\ + [\begin{matrix} 0 & X_{l o n g} & 0 & 0 \\ Y_{l a t} & 0 & 0 & 0 \\ 0 & 0 & 0 & Z_{t h r} \\ L_{l a t} & 0 & 0 & 0 \\ 0 & M_{l o n g} & 0 & 0 \\ 0 & 0 & N_{d i r} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] (\begin{matrix} δ_{l a t} \\ δ_{l o n g} \\ δ_{d i r} \\ δ_{t h r} \end{matrix}) \end{matrix}$ (2)

Closed-loop state space formulation

PID feedback was used to augment the vehicle’s stability, so the feedback controller was represented in state space form to calculate the vehicle’s closed-loop response. Reference⁴³ was used to formulate the closed-loop state space representation of a proportional-integral-derivative controller. Figure 13 shows the lateral and longitudinal feedback diagrams. The feedback control for both axes incorporated a cascaded PID control structure where the inner loop controls the body axis rates and the outer loop controls the attitude angles. Since longitudinal, lateral, and directional vehicle dynamics were later found to be decoupled, the formulation was performed separately for each axis. The formulation for the longitudinal axis is shown below. The procedure for the lateral and directional axes followed the same approach. $\begin{matrix} (\begin{matrix} \dot{u} \\ \dot{q} \\ \dot{θ} \end{matrix}) = [\begin{matrix} X_{u} & X_{q} & - g \\ M_{u} & M_{q} & 0 \\ 0 & 1 & 0 \end{matrix}] (\begin{matrix} u \\ q \\ θ \end{matrix}) + (\begin{matrix} X_{l o n g} \\ M_{l o n g} \\ 0 \end{matrix}) (\begin{matrix} δ_{l o n g} \end{matrix}) \end{matrix}$ (3)

The closed-loop formulation begins with the same formulation as the open-loop state space models, but the feedback controls are initially incorporated in the control input vector. The reduced state space equation for the longitudinal axis is shown in Eq. 3. Figure 13a shows the longitudinal feedback control diagram. In the longitudinal axis, the pilot commands a desired pitch attitude angle, $θ_{d e s i r e d}$ or the setpoint. The error, $e_{1} (t)$ is the difference between $θ_{d e s i r e d}$ and $θ$ where $θ$ is the estimated pitch attitude angle. $e_{1} (t) = θ_{d e s i r e d} (t) - θ (t)$ (4)

The outer loop controller consisted only of a proportional controller with the gain $K_{p_{1}}$ . The output follows the following form: $u_{1} (t) = K_{p_{1}} (θ_{d e s i r e d} (t) - θ (t)) = K_{p_{1}} e_{1} (t)$ (5)

The output of the previous equation is effectively the desired pitch rate, ${\dot{θ}}_{d e s i r e d}$ , and this is the input of the inner loop controller. The inner loop PID controller was implemented in parallel form, where the control output is the sum of the proportional ( $u_{2_{p}}$ ), integral ( $u_{2_{i}}$ ), and derivative ( $u_{2_{d}}$ ) terms, and the gains are given by $K_{p_{2}}$ , $K_{i}$ , and $K_{d}$ , respectively. $\begin{matrix} u_{2} (t) = u_{2_{p}} (t) + u_{2_{i}} (t) + u_{2_{d}} (t) = K_{p_{2}} e_{2} (t) \\ + K_{i} \int_{0}^{t} e_{2} (τ) d τ + K_{d} \dot{e_{2}} (t) \end{matrix}$ (6)

The rate error, $e_{2} (t)$ , is calculated between the desired pitch rate and the measured pitch rate, $\dot{θ}$ . $e_{2} (t) = {\dot{θ}}_{d e s i r e d} (t) - \dot{θ} (t)$ (7)

The proportional term of the inner loop is multiplied by the rate error. $u_{2_{p}} (t) = K_{p_{2}} ({\dot{θ}}_{d e s i r e d} (t) - \dot{θ} (t)) = K_{p_{2}} e_{2} (t)$ (8)

For the integral term, a new state variable, z, is defined to represent the integral action in terms of the rate error. $\begin{matrix} \dot{z} (t) = e_{2} (t) \\ u_{2_{i}} (t) = K_{i} z (t) \end{matrix}$ (9)

The derivative term is defined using the derivative of the pitch rate or the angular pitch acceleration instead of the derivative of the error rate. This avoids discontinuities in the feedback when the setpoint suddenly changes. $u_{2_{d}} (t) = K_{d} \dot{q}$ (10)

The expression for the $\dot{q}$ state from the reduced state space form (Eq. 3) substituted into the derivative term, and the simplification is as follows: $\begin{aligned} u_{2_{d}} (t) & = K_{d} \dot{q} \\ u_{2_{d}} (t) & = K_{d} (M_{u} u + M_{q} q + M_{l o n g} u_{2_{d}} (t)) \\ (1 - M_{l o n g} K_{d}) & u_{2_{d}} (t) = K_{d} (M_{u} u + M_{q} q) \\ u_{2_{d}} (t) & = \frac{K_{d} M_{u}}{1 - M_{l o n g} K_{d}} u + \frac{K_{d} M_{q}}{1 - M_{l o n g} K_{d}} q \end{aligned}$ (11)With the expressions for each term of the PID controller in terms of the state vector and the setpoint ( $θ_{d e s i r e d}$ ), the state space form of the closed-loop response can be assembled. $\begin{matrix} (\begin{matrix} \dot{u} \\ \dot{q} \\ \dot{θ} \\ \dot{z} \end{matrix}) = [\begin{matrix} X_{u} & X_{q} & - g & 0 \\ M_{u} & M_{q} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & - 1 & - K_{p_{1}} & 0 \end{matrix}] (\begin{matrix} u \\ q \\ θ \\ z \end{matrix}) \\ + [\begin{matrix} X_{l o n g} & 0 \\ M_{l o n g} & 0 \\ 0 & 0 \\ 0 & K_{p_{1}} \end{matrix}] (\begin{matrix} u_{2_{p}} + u_{2_{i}} + u_{2_{d}} \\ θ_{d e s i r e d} \end{matrix}) \end{matrix}$ (12) $\begin{matrix} (\begin{matrix} u_{2_{p}} + u_{2_{i}} + u_{2_{d}} \\ θ_{d e s i r e d} \end{matrix}) = \\ [\begin{matrix} K_{a} M_{u} & K_{a} M_{q} - K_{p_{2}} & - K_{p_{1}} K_{p_{1}} & K_{i} \\ 0 & 0 & 0 & 0 \end{matrix}] (\begin{matrix} u \\ q \\ θ \\ z \end{matrix}) \\ + [\begin{matrix} K_{p_{1}} K_{p_{2}} \\ 1 \end{matrix}] (\begin{matrix} θ_{d e s i r e d} \end{matrix}) \end{matrix}$ (13)where $K_{a} = \frac{K_{d}}{1 - M_{l o n g} K_{d}}$

The terms of the control input vector can be combined with the open-loop A matrix to get the closed-loop A matrix, $A_{c l}$ . The remaining terms are regrouped in the closed-loop $B_{c l}$ matrix, and the new control input vector is reduced to $u_{c l} u_{c l} = [θ_{d e s i r e d}]$ . $\begin{matrix} A_{c l} = [\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ 0 & 1 & 0 & 0 \\ 0 & - 1 & - K_{p_{1}} & 0 \end{matrix}] \\ B_{c l} = [\begin{matrix} X_{l o n g} K_{p_{1}} K_{p_{2}} \\ M_{l o n g} K_{p_{1}} K_{p_{2}} \\ 0 \\ K_{p_{1}} \end{matrix}] \end{matrix}$ (14)where $\begin{aligned} a_{11} & = X_{u} + X_{l o n g} K_{a} M_{u} \\ a_{12} & = X_{q} + X_{l o n g} (K_{a} M_{q} - K_{p_{2}}) \\ a_{13} & = - g - X_{l o n g} K_{p_{1}} K_{p_{2}} \\ a_{14} & = X_{l o n g} K_{i} \\ a_{21} & = M_{u} + M_{l o n g} K_{a} M_{u} \\ a_{22} & = M_{q} + M_{l o n g} (K_{a} M_{q} - K_{p_{2}}) \\ a_{23} & = - M_{l o n g} K_{p_{1}} K_{p_{2}} \\ a_{24} & = M_{l o n g} K_{i} \end{aligned}$

Flight testing

To estimate the model coefficients, flight tests were performed to systematically excite each of the vehicle states, and each of the states (x) and inputs (u) were measured. The interrogation volume of the space was 15 ft (length) x 15 ft (width)x 8 ft (height). A set of six infrared (Vicon) motion capture cameras was used to track the position of the retro-reflective markers placed on the vehicle. The vehicle in the motion capture space is shown in Figure 11. With the position of the reflective markers with respect to the CG of the vehicle known, the Vicon system can determine the inertial position and the attitude of the vehicle using Euler parameters (quaternions) during flight testing. The Euler parameters were converted to roll ( $ϕ$ ), pitch ( $θ$ ), and yaw ( $ψ$ ) Euler angles and body angular velocities after flight testing. The Euler angles and body angular velocities estimated by the Vicon system were compared against the telemetry transmitted from the autopilot’s inertial measurement unit (IMU), and there was good agreement between the two independent state measurements. Additionally, the final control outputs were also included in the telemetry data from the autopilot. All of the states and control inputs from both Vicon and the autopilot were recorded at a ground station at a rate of 40 Hz.

The flight experiments were divided into three categories: lateral and longitudinal experiments, directional experiments, and heave experiments. This was based on previous observations of the system’s behaviors to different pilot inputs. Figure 14 demonstrates the type of excitation given for each set of experiments. Since the doublet excitations were manually executed by the pilot, enough variation was created in the measured output to ensure adequate excitation of the system for parameter estimation. Only longitudinal and lateral coefficients were determined from the separate pitch and roll excitation data sets. This ensured a good signal-to-noise ratio for a higher degree of confidence in the resulting coefficients. Similarly, directional and heave experiments were performed separately from each other. For each category of flight test experiments, over 50 excitations were given. The individual parameters were selected and averaged based on criteria discussed in the Results section. The results of the different types of flight experiments were combined into a single flight dynamics model.

Figure 11.

TLMAV in Vicon motion capture space.

Methodology for exciting longitudinal and lateral dynamics

The first set of systematic flight experiments targeted the vehicle’s longitudinal and lateral dynamic modes because they appeared to be decoupled from directional and heave dynamics. Since the current vehicle design is approximately axisymmetric, the model coefficients for the pitch and roll were expected to be similar in value. For these experiments, the pitch and roll gains were reduced as much as possible while ensuring that the pilot could still control the vehicle to obtain the near open-loop vehicle response. The yaw gains remained at the values determined from previous flight testing since the yaw axis should be decoupled from the present axes of interest.

The vehicle was brought to a stable hover in the motion capture space, and the longitudinal and lateral dynamics were excited by the pilot’s commanding pitch and roll doublets. Each excitation was provided for a single axis, and the vehicle returned to hovering flight before the next excitation input was applied. During each flight experiment, 10 to 20 separate excitations could be performed over a 6 to 8 minute period.

Figure 12.

Body reference frame used for system identification.

Figure 13.

Lateral and longitudinal cascaded feedback control diagrams. (a) Longitudinal feedback diagram; (b) Lateral feedback diagram.

Figure 14.

Categories of flight experiments performed to extract the flight dynamics model. (a) Lateral and longitudinal experiments: excitation given by tilting thrust vector ( $δ_{l a t}$ or $δ_{l o n g}$ ); (b) Directional experiments: excitation given by generating differential rotor torque ( $δ_{d i r}$ ); (c) Heave experiments: excitation given by varying rotor thrust from hover ( $δ_{t h r}$ ).

Methodology for exciting directional and heave dynamics

The directional dynamics of the vehicle were targeted in the second set of flight experiments. To prevent off-axis disturbances, the pitch and roll gains were increased to allow the pilot to easily command yaw excitations with minimal drift. The yaw gains were reduced enough to prevent the vehicle from oscillating. Since the directional axis has the smallest moment of inertia of the three principal axes, only small inputs are required to yaw the vehicle. This also means that very small gains are needed to stabilize yaw compared to pitch and roll. To adequately perturb the yaw axis, the desired yaw rate was set to zero, and the pilot yaw input was added to the output of the yaw control loop. This was done because the reduced gains prevented the vehicle from achieving the high yaw rates needed to increase the signal-to-noise ratio for model identification. As before, the vehicle was brought to a stable hovering state in the motion capture space, and a yaw disturbance was commanded. The vehicle was then allowed to return to hovering flight before the next excitation was provided.

For the final set of experiments, the heave dynamics were targeted. Similar to the yaw experiments, the pitch and roll gains were increased to prevent drift, and the yaw gains remained low. After the vehicle obtained a hovering state, a throttle doublet was commanded, causing the vehicle to accelerate upwards and then descend before returning to hover.

Time history data

During the longitudinal and lateral flight dynamics experiments, it was observed that the off-axis response was negligible. This confirmed the previous hypothesis that this vehicle’s longitudinal and lateral flight dynamic modes are decoupled from each other. The time history from a representative set of flight test data, including longitudinal and lateral excitations, are shown in Figs. 15 and 16, respectively, including the control inputs and vehicle states. The cross coupling coefficients such as $X_{v}$ , $X_{p}$ , $Y_{u}$ , $Y_{q}$ , $L_{q}$ , and $M_{p}$ were not included in the model. An attempt was made to include these other terms in the model, but correlations were insufficient to include them in the final model structure. Note the magnitudes of the excitations of these six states are significant: a maximum translational velocity of 2 m/s (6.5 ft/s) and a rotational velocity of 5 rad/s (290 deg/s) were generated. These disturbances were large enough to generate a model for the longitudinal and lateral states.

Figure 15.

Time history of longitudinal data from a pitch excitation.

Figure 16.

Time history of lateral data from a roll excitation.

The directional flight experiments were performed separately from the other flight experiments. This was done primarily because the vertical axis is very sensitive to a yaw control input, and it was difficult to provide a pure yaw input without also imparting a heave input. The time history from a sample set of flight test data is shown in Figure 17, which includes the control input and vehicle states. Here, the size of the control input is less than 10% of the maximum possible inputs, and the vehicle can achieve yaw rates up to 10 rad/s (600 deg/s) or more within a quarter of a second. Compared to the pitch and roll rates, large yaw rate excitations were needed to identify coefficients for the $\dot{r}$ and $\dot{ψ}$ models.

Figure 17.

Time history of directional data from a yaw excitation.

Finally, the heave flight experiments were conducted to complete the linear model of the vehicle. The time history from a sample set of flight test data is shown in Figure 18, which includes the throttle input and w body velocity. For these experiments, the maximum heave translational velocity attained was 2 m/s (6.5 ft/s), sufficient to extract a model for $\dot{w}$ .

Figure 18.

Time history of heave data from a throttle excitation.

Using the flight test data from each type of experiment, the coefficients of the A and B matrices were determined from the least squares solutions using $\dot{x}$ as the output and a combination of the measured states and control signals as inputs. The method of step-wise regression was used to determine the linear model structure for each output state. In this framework, each input state is termed a regressor, and the statistical correlation between the output and the regressor is calculated. Only regressors with a significant correlation with the output are included in each model. Two statistical metrics were used to determine the significance of each regressor: the partial F-ratio and the coefficient of determination ( $R^{2}$ ). An F-ratio of at least 20 corresponds to a confidence level of 95%. Regressors not included in the postulated model were tested, and the regressors with F-ratios less than 20 were removed from the model. This meant the regressors did not have a significant enough correlation with output to be included in the identified model. In addition to the F-ratio threshold, coefficients from individual datasets with $R^{2}$ values less than 75% were not included, with the average used to determine the value of each coefficient. These statistical metrics are incorporated in the SIDPAC software package.¹⁷

The values of each coefficient were averaged over all of the datasets from the flight experiments. The numerical value of each resulting coefficient is presented in Table 2. The standard errors as a percentage of the coefficient values are also shown in the table. The standard errors for these coefficients are less than 15%, which means the uncertainty in the experimentally derived parameters is relatively small.

Table 2.

Experimentally derived linear model parameters for the TLMAV.

Parameter	Value	Standard error, %
Longitudinal mode
$X_{u}$ , 1/s	−0.299	7.2
$M_{u}$ , rad/m/s	6.77	3.3
$X_{q}$ , m/s/rad	−0.165	5.6
$M_{q}$ , 1/s	1.42	14.0
$X_{l o n g}$ , m/s²	3.95	2.6
$M_{l o n g}$ , rad/s²	−61.8	1.0
Lateral mode
$Y_{v}$ , 1/s	−0.250	11.0
$L_{v}$ , rad/m/s	−7.08	4.6
$Y_{p}$ , m/s/rad	0.173	4.6
$L_{p}$ , 1/s	1.54	14.0
$Y_{l a t}$ , m/s²	3.75	2.5
$L_{l a t}$ , rad/s²	66.1	0.9
Directional mode
$N_{r}$ , 1/s	−2.84	8.5
$N_{d i r}$ , rad/s²	−1430	2.0
Heave mode
$Z_{w}$ , 1/s	−0.353	3.1
$Z_{t h r}$ , m/s²	−14.3	0.8

To verify the resulting model, The derived state-space model was compared against independent measured data sets. The $R^{2}$ values were calculated for each state of the model. These results and $R^{2}$ values are shown in Figs. 19–22. The comparison of the measured and modeled longitudinal states $\dot{u}$ , $\dot{θ}$ , and $\dot{q}$ are shown in Figure 19, and the corresponding comparison of lateral states are shown in Figure 20. The comparison of the measured and modeled directional states $\dot{ψ}$ and $\dot{r}$ are shown in Figure 21, and the corresponding comparison of the heave state is shown in Figure 22. For each model, the $R^{2}$ values were between 84% and 100%, indicating that the model agrees well with the experimental data.

Figure 19.

Comparison between experimental and modeled outputs for longitudinal dynamics.

Figure 20.

Comparison between experimental and modeled outputs for lateral dynamics.

Figure 21.

Comparison between experimental and modeled outputs for directional dynamics.

Figure 22.

Comparison between experimental and modeled outputs for directional dynamics.

Lateral and longitudinal modes

The identified stability and control derivatives for the longitudinal and lateral modes are given in Table 2. The values of the coefficients for the longitudinal and lateral modes are nearly identical, which is reasonable given the axisymmetric nature of the vehicle. However, there are some differences in signs, which can be attributed to the definition of the body-fixed frame.

To better interpret meanings of the lateral and longitudinal stability derivatives, Figure 23 shows the response to a u and v translational velocity perturbation. The negative sign on the $X_{u}$ and $Y_{v}$ terms suggests that both are stabilizing. In the presence of a velocity perturbation in either the X or Y direction, a stabilizing drag force from the vehicle fuselage and rotors acting above the CG will tend to mitigate the perturbation. The same disturbance will also cause a nose-up (positive) moment in the case of a positive u disturbance or a roll left (negative) moment in the case of a positive v disturbance, according to the signs of $M_{u}$ and $L_{v}$ . This will cause the vehicle to rotate about the CG in the direction of the disturbance.

Figure 23.

Interpretation of the lateral and longitudinal stability derivatives. (a) Longitudinal response to positive u perturbation; (b) Lateral reponse to positive v perturbation.

The signs of both $M_{q}$ and $L_{p}$ are positive, which indicates that the pitch and roll rate damping terms make the system unstable. The unstable angular rate damping terms, in combination with the larger $M_{u}$ and $L_{v}$ terms, create an unstable oscillatory mode. This is likely because the center of pressure (CP) for the fuselage, in the wake of the rotors, is above the CG, as shown in Figure 23. The moment about the CG due to translational velocity has no natural damping, so the rotation continues unabated. This behavior is why pitch and roll feedback is necessary to stabilize the vehicle in flight.

The vehicle response to positive lateral and longitudinal control inputs is shown in Figure 24. A longitudinal control input, $δ_{l o n g}$ , imparts an angular acceleration, $M_{l o n g}$ , due to the moment generated by displacing the thrust vector away from the CG and a translation acceleration, $X_{l o n g}$ , will develop as the component of the thrust is along the x-axis. An analogous response will occur with a lateral control input, $δ_{l a t}$ . It was calculated from the value of $M_{l o n g}$ and $L_{l a t}$ that the maximum pitch or roll angular acceleration was between 62 and 66 rad/s². The maximum translational acceleration based on $X_{l o n g}$ and $Y_{l a t}$ was nearly 4 m/s² (13 ft/s²).

Figure 24.

Interpretation of the lateral and longitudinal control derivatives. (a) Lateral response to positive control input, $δ_{l o n g}$ ; (b) Longitutdinal reponse to positive control input, $δ_{l a t}$ .

Directional mode

The identified stability and control derivatives for the directional mode are given in Table 2. The negative sign on the yaw damping term $N_{r}$ suggests that both are stabilizing. Of all the control derivatives, the yaw control derivative $N_{d i r}$ is the largest. Figure 25a shows the directional response to a yaw input, $δ_{d i r}$ . This is primarily because the yaw inertia of the vehicle is significantly smaller compared to the other two axes. For a unit yaw control input, the maximum yaw angular acceleration of the vehicle is 1,430 rad/s², which is the largest possible angular acceleration among roll, pitch, and yaw. During a typical flight test, the yaw control input rarely exceeds 3% of its maximum value. The extreme sensitivity of the yaw control revealed the need for a small amount of feedback control to dampen the pilot’s yaw control inputs during normal flight.

Figure 25.

Directional and Heaves reponse to control inputs. (a) Directional response to positive control input, $δ_{d i r}$ ; (b) Heave reponse to positive control input, $δ_{t h r}$ .

Heave mode

The identified stability and control derivatives for the directional mode are given in Table 2. The heave mode is decoupled from the other modes, and only $Z_{w}$ and $Z_{t h r}$ terms could be identified. The negative sign of the $Z_{w}$ implies that the heave mode is stable. Figure 26 shows the heave response to w perturbation. The coefficient $Z_{w}$ likely corresponds to the acceleration due to drag acting on the fuselage and rotors, which explains the damping effect of the coefficient. Figure 25(b) shows the heave response to a thrust input, $δ_{t h r}$ . It was calculated from the value of $Z_{t h r}$ that the maximum heave acceleration was 14 m/s² (46 ft/s²).

Figure 26.

Heave response to w perturbation.

Eigenstates

The eigenvalues of the open-loop state transition matrix from Table 2 were computed. The resulting eigenvalues are shown in Table 3 and Figure27. The damping ratio, $ζ$ , and the natural frequency, $ω_{n}$ , are also included. Both the longitudinal and lateral modes have two sets of nearly identical eigenvalues, which is expected for an axisymmetric body with the axis of symmetry along the vertical axis. One is stable and critically damped, while the other is undamped and unstable ( $ζ < 0$ ) and oscillatory. The unstable modes demonstrate the need for feedback control to stabilize the vehicle in flight. Both the directional and heave modes are critically damped. An additional eigenvalue exists at the origin due to the yaw angle being included as a state; however, it is not shown.

Figure 27.

Open-loop eigenvalues for TLMAV in hover.

Table 3.

Open-loop eigenvalues for TLMAV in hover.

Mode	Eigenvalues	$ζ$	$ω_{n}$ , rad/s
Longitudinal: $u, q, θ$	−3.67
Longitudinal: $u, q, θ$	2.43±3.56i	−0.56	4.31
Lateral: $v, p, ϕ$	−3.66
Lateral: $v, p, ϕ$	2.48±3.66i	−0.57	4.35
Directional: $r, ψ$	−2.84
Heave: w	−0.353

The lateral and longitudinal closed-loop eigenvalues were calculated using the previously identified coefficients and the controller gains that were used for free flight. Specifically, the cascaded PID controller employed an outer-loop proportional gain of $K_{p 1} = 3.50$ , and inner-loop gains of $K_{p 2} = - 0.180$ , $K i = - 1.47 * 10^{- 4}$ , and $K_{d} = 1.64 * 10^{- 4}$ . Figure 28 compares the open-loop and closed-loop eigenvalues for lateral and longitudinal modes. The closed-loop eigenvalues are shown in Table 4. The feedback controller shifts the two previously identified unstable pole pairs from the right half-plane to the left half-plane. This means that the unstable models from the loop-loop model are now stable and damped with the addition of the feedback controller. The damping ratios for the previously unstable modes became positive, indicating that the controller provides a significant increase in damping to the oscillatory response of the system. The key insight is that the same controller gains can stabilize both the mathematical and physical models of the vehicle, further demonstrating the validity of the extracted model and feedback structure.

Figure 28.

Comparison of the open-loop and closed-loop eigenvalues for lateral and longitudinal modes of the TLMAV in hover.

Table 4.

Closed-loop eigenvalues for TLMAV in hover.

Mode	Eigenvalues	$ζ$	$ω_{n}$ , rad/s
Longitudinal: $u, q, θ$	−4.297
Longitudinal: $u, q, θ$	−2.858±2.295i	0.78	3.67
Lateral: $v, p, ϕ$	−4.589
Lateral: $v, p, ϕ$	−3.058±1.804i	0.86	3.55

Summary and conclusions

In this study, the flight dynamics of a novel coaxial helicopter MAV with a thrust vectoring mechanism for pitch and roll control were investigated using flight test data. This type of vehicle is more compact, efficient, and quieter than a multirotor configuration and also represents a mechanically simpler alternative to traditional coaxial configurations with swashplate-based control. The vehicle is designed in such a way that it could potentially be launched from a grenade launcher to extend the range and minimize response time. The objective of the flight experiments was to identify the open-loop response of the vehicle and to develop a linear time-invariant flight dynamics model in hovering flight. The cascaded PID feedback controller implemented on the lateral and longitudinal axes was represented in the state space form. Systematic flight tests were performed to excite the system’s longitudinal, directional, and heave dynamic modes. The least-squares approach based on SIDPAC was used to extract an open-loop system model using vehicle states and controls as inputs and the derivative of vehicle states as output. The results included the stability and control derivatives for the A and B state-space matrices. The derived state-space model was compared with independent measured data sets. The $R^{2}$ values of the identified model were greater than 80% for each vehicle state. The eigenvalues of the open-loop and closed-loop state transition matrices were calculated. The identified LTI model can be used as the basis for establishing a non-linear model.

The key conclusions are enumerated below:

Longitudinal and lateral modes are decoupled and symmetric. The longitudinal and lateral dynamics were found to be decoupled from each other and from the heave and directional modes, consistent with the vehicle’s near-axisymmetric design. Stability and control derivatives in both axes were similar in magnitude. The translational damping terms ( $X_{u}$ and $Y_{v}$ ) were negative and thus stabilizing, while the rotational damping terms ( $M_{q}$ and $L_{p}$ ) were positive and destabilizing. This combination produced one stable, critically damped mode and one unstable oscillatory mode in each axis. The presence of these unstable modes confirms the need for feedback control to maintain stable pitch and roll control.

Heave and directional modes are decoupled, with high sensitivity and control effectiveness in yaw. The heave and direction dynamics were found to be decoupled from each other. Both damping terms are negative and, therefore, stable and damped. The large $N_{r u d}$ term, as compared to the other control coefficients, means that the vehicle can achieve extremely high yaw angular accelerations. This is due to the low moment of inertia about the vertical axis. This extreme yaw sensitivity reveals that small feedback gains are necessary to dampen the pilot’s yaw control inputs.

PID controller stabilized both the linear and physical vehicle models. The cascaded feedback controller implemented on the lateral and longitudinal axes stabilized the open-loop unstable modes of the linear mathematical model. The damping on the previously unstable models was increased. The same controller gains also stabilized the physical vehicle.

The dual-swashplate mechanism for the cyclic pitch control of a coaxial rotor system is extremely complex and adds significant drag in forward flight. These are the reasons why there are not many coaxial helicopters in existence today. The current study sheds light on the flight dynamics of a novel coaxial rotorcraft that uses rotor-head tilting, which is mechanically much simpler than cyclic blade pitching via two swashplates for pitch and roll control. This has broader impact beyond the current TLMAV concept, such as larger UAVs and even full-scale VTOL aircraft.

Footnotes

Acknowledgements

The authors would like to thank Dr. Vikram Hrishikeshavan for his assistance with programming the ELKA-R autopilot. The authors would also like to thank Dr. Manoranjan Majji for allowing us to use the Vicon® motion capture space at the Land,Air,and Space Robotics (LASR) Laboratory to collect flight testing data.

ORCID iD

Hunter Denton

Moble Benedict

Funding

The authors disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Numbers W911NF-19-2-0057 and W911NF-21-2-0188. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies,either expressed or implied,of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes,notwithstanding any copyright notation herein.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

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