Design and Flight Dynamics of a Hand-Launched Foldable Micro Air Vehicle ^†

Abstract

This paper discusses the development, flight-testing, and flight dynamics modeling of a Micro Air Vehicle (MAV) that could be deployed in a folded configuration via hand launching. This 112 g MAV features foldable propeller arms that can lock into a compact rectangular profile comparable to the size of a smartphone. The vehicle can be launched by simply throwing it in the air, at which point the arms would unfold and autonomously stabilize to a hovering state. Multiple flight tests demonstrated the capability of the feedback controller to stabilize the MAV from different initial conditions including tumbling rates of up to 2500 deg/s. A six-degree-of-freedom flight dynamics model was developed and validated using flight test data obtained from a motion capture system for various hand-launched scenarios. The current MAV, with its compact design, extreme portability, and rapid/robust deployment capability, could be ideal for emergency scenarios, where a standard launch procedure is unfeasible.

1. Introduction

Micro Air Vehicles (MAVs) are a specific class of Uncrewed Aerial Vehicles (UAVs) that were originally defined by DARPA in 1995 to be aircraft with maximum dimensions (length, width, or height) smaller than 15 cm, a mass of 100 g or less with a payload of 20 g, and an endurance of an hour [1]. Unfortunately, the small size, higher power requirements for vertical flight, aerodynamic inefficiencies at low Reynolds numbers, and low energy density of batteries severely limit the endurance of hover-capable MAVs to typically less than 20 min [2,3,4]. Despite their endurance limitations, their portability, low cost, and ease of deployment make MAVs extremely valuable for civilian and military applications. They play a critical role in a variety of applications, such as intelligence, surveillance, and reconnaissance (ISR), as well as search and rescue missions, because they provide robust platforms for real-time video and sensor feeds to the ground operator.

The small size and high maneuverability of hover-capable rotary-wing MAVs make them a promising solution for ISR in crowded urban environments. For example, researchers at ETH Zürich have developed navigation software for MAVs to be extremely effective in search and rescue missions, specifically allowing them to fly through tight spaces caused by earthquake damage [5]. Although there have been many studies to understand MAV flight dynamics and control, navigation, aerodynamics, aeroelasticity, etc., an area that has lacked attention is the robust deployability of MAVs [6,7,8,9,10,11,12,13]. There could be emergency scenarios where the MAV has to be deployed rapidly in the field and therefore taking off from a stationary surface may not be feasible. For example, there could be a need for rapid ISR on the battlefield, and the warfighter may have to take out a folded MAV and simply throw it up in the air to deploy it. This means that the MAV has to first self-deploy (or unfold) in the air, then stabilize itself from a tumbling state to a trimmed state, and further perform the mission. Such scenarios form the motivation for the present work.

1.1. Previous Work

The proposed hand-launched MAV falls under the broad umbrella of air-launched concepts. Although there have been some air-launched aircraft in the past, the authors are not aware of any previous work on hand-launched foldable MAVs. AeroVironment has demonstrated the potential of hand-launched aircraft through Raven B RQ-11, which is the most widely deployed Uncrewed Aerial System (UAS) in the world [14]. Since this aircraft could be hand-launched, it serves as a rapidly deployable ISR UAS. One of the key shortcomings of this aircraft is that while it is portable inside a military unit’s backpack, it adds an additional 2 kg to be carried by the military personnel and has to be assembled onsite, thus increasing the warfighter’s load and reducing the agility of deployment during emergencies.

Although not an air-launched concept, a UAS that demonstrates the advantages of an ultra-compact MAV is the Black Hornet Nano by FLIR [15]. Black Hornet is utilized in covert ISR missions; it is equipped with thermal cameras, has a range of one mile, and a total system weight of 1.2 kg (although the Black Hornet MAV itself only weighs 18 g). The device follows the typical launch procedure of other vertical take-off and landing (VTOL) UAS, thus reducing the robustness of deployment and increasing the time it takes for the vehicle to get airborne.

The Raven B RQ-11 showcases the true potential of hand-launched vehicles, as they can have an immediate aerial vantage point in ISR missions soon after they are launched. On the other hand, the Black Hornet Nano is a testament to the effectiveness of an ultra-compact MAV and the significance of portability. Therefore, an aircraft that is inherently portable by being ultra-compact, but still has the same rapid-deployment advantages of a hand-launched system, becomes an interesting research opportunity. This paper presents the development and testing of such a vehicle.

Another genre of rapidly deployable air-launched aircraft that has recently received significant attention is tube-launched UAS. A notable example of this is the Streamlined Quick Unfolding Investigation Drone (SQUID), which was developed at Caltech [16]. This foldable, ballistically launched drone does not fit within the DARPA-defined original size restrictions of an MAV—the length is 79 cm, the unfolded diameter is 58 cm, and the weight is 3.3 kg—but shows the advantages of foldable vehicles: it is compact with a diameter of only 15 cm when folded, and a launch has been demonstrated from a platform moving at a speed of 12 m/s. The SQUID folds by utilizing spring-loaded hinges which are not locked in place but rather restrained by the launcher itself. Neither the folding mechanism nor the control strategy of SQUID can be implemented directly in the vehicle described in this paper; however, it provides a useful starting point to design both aspects of this vehicle. Although SQUID demonstrates the utility of a folding UAS, it can only be launched from a purpose-built cannon, limiting the deployment and portability of the UAS due to the necessity of carrying the deployment device alongside the vehicle.

Another tube-launched example is the Gun-Launched UAS (GLUAS) developed at Texas A&M University [17]. GLUAS utilizes a coaxial rotor system rather than an unfolding mechanism with rotor arms like the SQUID to make it better suited to fit inside a grenade launcher already carried by military personnel. This aircraft can be ballistically launched directly to its final desired hovering location, increasing the range of the device by eliminating the necessity to take off and fly to that location and expending battery energy in the process. While GLUAS can be launched from existing hardware, it still requires the grenade launcher to be carried along with the vehicle to be deployed.

Although the current MAV is hand-launched, it faces many of the same challenges as tube-launched MAVs, which primarily include (1) a mechanism to fold the MAV into a more compact state and automatically unfold after launching, and (2) a control strategy to stabilize the aircraft from a tumbling state. Although a hand-launched device does not have the battery-energy-saving advantages of the GLUAS, where the UAS is launched as a projectile to its final desired location, it capitalizes on compactness and the ability to rapidly deploy the MAV without requiring any assembly or additional hardware. Simply put, the current MAV can be removed from a pocket and thrown into the air and will perform the mission without any human involvement.

1.2. Current Concept

The current work intends to advance the state of the art of MAVs by developing an extremely portable, foldable, compact (form factor of a smartphone once folded) and hand-launched quadcopter designed for ISR and search and rescue missions. The MAV can be hand-launched by throwing it in the air, where the arms and rotors deploy at the apex of the trajectory and the aircraft autonomously stabilizes. A sequenced image of a hand-launched flight test showcasing the flight profile is shown in Figure 1. A flight test video can be found in [18]. The foldability and launch style make the device extremely portable (it can fit in any standard pocket) and rapidly deployable without requiring any specialized launch equipment. Once deployed, it can immediately provide an aerial vantage point. This type of vehicle has military and civilian applications. For example, the MAV could be equipped with a camera and be carried in the pockets of military personnel to be used as an instantly deployable surveillance UAS. Similarly, this UAS could be used as a mobile security system for civilians, where they could carry it in their pocket/purse and rapidly deploy it in areas where they felt unsafe; the UAS would then be able to track the deployer and livestream a video of what is happening to an emergency responder. However, in order to develop such a robust MAV, the following barriers would need to be overcome: (1) design and build a compact quadcopter with foldable arms that fits within the footprint of a smartphone, (2) design a mechanism to automatically deploy the rotors at the launch apex, and (3) develop a control strategy to autonomously recover the vehicle after launch from arbitrary initial conditions.

The present paper is an extension of a conference paper published in Vertical Flight Society 80th Annual Forum & Technology Display [19]. The paper discusses the design of this aircraft, the implementation of the control methodology on a 7 g flight controller, the demonstration of the proof of concept via flight testing, the measurement of vehicle dynamics via a three-dimensional (3D) motion capture system (Vicon^®), the development of a six-degree-of-freedom (6-DOF) flight dynamics model, and validation of the model using flight test data.

2. Vehicle Design

Currently, an MAV with its maximum dimension, length, of approximately 15 cm when unfolded and approximately 11 cm when folded is designed and built (shown below in Figure 2). The width of the vehicle when unfolded is approximately 12 cm and approximately 6 cm when folded. The main body of the MAV is 3D-printed using PLA plastic and provides anchor points for the rotatable propeller arms, servo motor, and flight controller. The propeller arms and flight controller mounts are also 3D-printed with PLA plastic. Each propeller arm can independently rotate about a steel pin mounted to the main body and is restricted from rotating using a torsional spring. The moment of the spring maintains the propeller arm in the unfolded position, as seen in Figure 2a. The flight controller mount serves a dual purpose because it also restrains the 2-cell (7.4 V) Lithium Polymer (LiPo) battery beneath the flight controller. The propeller arms can be folded individually and once folded, the motors and propellers fit within the rectangular profile of the quadcopter, as shown in Figure 2b. The weight breakdown of the vehicle is shown in

Table 1. Weight breakdown of the MAV.

Components	Mass (g)	% Total
Propellers	2.0	1.79
Motors	20.0	17.86
Servo Motor	10.1	9.02
Release Arm	0.5	0.45
Propeller Arms	10.0	8.93
Fuselage	22.9	20.45
Flight Controller	6.6	5.89
Battery	18.2	16.25
Receiver	1.5	1.34
Wires and Screws	15.5	13.84
Vicon® Markers	4.7	4.0
Total	112.0 g	100.0%

. The center of mass of the vehicle is almost at the geometric center, however, shifted longitudinally towards the servo motor (rear of the quadcopter) by just 4.5 mm or roughly 3% of the length and with negligible variations from the center in the lateral direction. The battery accounts for 16.25% of the vehicle’s mass, which restricts the endurance. Specifically, since the battery has a capacity of 350 mAh and the quadcopter draws approximately 1.3 A in hover, the endurance is around 13 min when the LiPo battery is discharged by 80% (standard discharge amount for battery health). As stated in the Introduction, this vehicle also struggles with endurance like other MAVs; however, the vehicle consumes less power during take-off due to hand launch, and a higher-capacity battery can extend endurance.

2.1. Propeller Arm Folding/Deployment Mechanism

To keep the quadcopter in its folded position, a locking mechanism was designed that works by having an aluminum release arm attached to a servo motor (see Figure 3). The release arm locks the two propeller arms that are closer to the servo motor. The other two propeller arms are folded and held back within the rectangular profile by the two locked propeller arms. The folding of the propeller arm would wind the respective torsional spring mentioned before. When the servo motor is turned clockwise, the release arm no longer locks the propeller arms in the folded position, allowing them to unfold using the restoring torque of the wound torsional spring. Once the arms are fully unfolded, the propellers could spin up and stabilize the quadcopter. Betaflight software™, version 4.3.1, is used to program the flight controller in such a way that the two positions (locked and unlocked) of the servo motor can be controlled by flipping a switch on the RC transmitter. Furthermore, once the arms are unfolded, the controller automatically starts the motors and begins to stabilize the quadcopter from any initial orientation, velocity, and acceleration.

2.2. Control Methodology

The roll and pitch of the MAV are controlled using a cascaded proportional-integral-derivative (PID) feedback controller; the control architecture is shown in Figure 4a. The gyroscope on the flight controller measures the body axis rates up to ±250 deg/s, and the pitch, roll, and yaw attitudes are calculated from the flight controller’s three-axis accelerometer, which can measure accelerations up to ±2 g; the angular rate and attitude data are then passed through a low-pass filter to obtain a more accurate estimation of the actual MAV dynamics. The pilot commands a desired pitch and roll attitude with an RC transmitter, limited to ±35° and the error between the desired attitude and the actual attitude is scaled by the proportional-only outer loop controller to the desired rate. From there, the desired rate becomes the setpoint for the inner loop. The error between the setpoint and the actual angular rate is passed to the inner-loop PID controller. The PID computations are performed at a frequency of 4 KHz. Once both loops stabilize, the controller alters the revolutions per minute (RPM) of each motor to produce the desired attitude output for the MAV.

Instead of directly controlling the yaw angle, the yaw rate of the vehicle is controlled using only a PID controller; the control architecture is shown in Figure 4b. Therefore, the pilot commands the desired yaw rate with the RC transmitter instead. The error between the desired rate and the actual yaw rate, obtained by low-pass filtering the flight controller’s gyroscope data as before, is passed into the PID controller. Once the loop stabilizes, the output is supplied to the MAV by altering the RPM values of each motor to produce the desired yaw rate.

The vehicle was tuned through a standard trial and error method. The integral and derivative terms were set to zero while the proportional gain for each axis was increased until oscillations were observed in the output. At that point, the proportional gain was reduced by 10% and the integral gain was increased until the oscillations reappeared and then the term was reduced by 10%. With these new settings, vehicle dynamic modes were excited from a hovering state, and the proportional and integral terms were altered until the vehicle was stable and had a satisfactory response to pilot inputs. For the current concept, the controller has to quickly recover from aggressive tumbling and yet remain stable enough to not induce any other oscillations after recovery. Finally, a small derivative term was implemented to provide damping to the system. The derivative term was kept low to reduce noise amplification in the gyroscope; furthermore, a low-pass filter was implemented into the gyroscope at 500 Hz for the same reason. The final PID gain settings are shown in

Table 2. PID gain settings for the MAV.

	P	I	D
Roll	45.0	60.0	20.0
Pitch	47.0	62.0	20.0
Yaw	45.0	80.0	0.0

.

3. Testing

In order to test the capabilities of the proposed vehicle, two types of flight tests were performed: a proof-of-concept test and a motion capture test. The proof-of-concept test was conducted first, and it consisted primarily of throwing the folded MAV into the air, followed by the deployment of the propeller arm and a stable hover. The objective of this test was to prove that the vehicle performed as expected. The goal of the motion capture testing was to obtain more quantitative position and attitude data to understand the flight characteristics of the MAV and to validate a 6-DOF flight dynamics model. In addition to flight tests, the motor and propeller combination was also tested on a 6-DOF load cell to obtain thrust and torque as a function of rotational speed (RPM). The thrust and torque characteristics were used in the flight dynamic simulations.

3.1. Proof-of-Concept Testing

The first step in testing the MAV was ensuring that it would be able to recover from an arbitrary spinning state and stabilize itself. Toward this end, the vehicle was mounted on a single-degree-of-freedom test stand as shown in Figure 5. Once the controller was tuned to stabilize the vehicle on the stand from any arbitrary orientation and angular velocities, free flight tests were conducted in a flight cube, as shown in Figure 6. Next, the MAV was thrown in an already unfolded state within the flight cube in a benign fashion without intentionally inducing any initial angular rates. Then, the same test was repeated; however, this time, the vehicle was in a folded state. The final step was to demonstrate that the vehicle can recover from being thrown aggressively in a folded state where high angular rates were deliberately induced during the throw. These tests were conducted outdoors due to the lack of space in the flight cube. Numerous folded tests were performed, inducing different magnitudes of angular rates while hand-launching; one of which can be seen in Figure 1, where the MAV tumbled three times before stabilizing itself (video provided in [18]). The culmination of all of these tests proved that the current vehicle meets the objectives of the device.

3.2. Flight Testing in Motion Capture System

After the concept was proven, the next goal was to perform similar tests in a 3D motion capture environment (Vicon^® system) to obtain the inertial position, velocity, and acceleration data to validate a 6-DOF flight dynamic simulation of the hand-launched MAV. The Vicon^® Motion Capture system, Vicon^® Tracker 3.10, works by installing reflective markers on the vehicle and mapping them to an object in the software. When infrared cameras pick up at least four markers, they can track the object and estimate its Euler angles, quaternions, and position in 3D space. For the present study, eight cameras were used to track the position of Vicon^® markers on the vehicle in a control volume (Figure 7); therefore, all of the tests were performed in a windless environment. The Vicon^® marker layout utilized to track the MAV body can be seen in Figure 8. Four markers are placed in the same plane and separated by at least 10 cm. A fifth marker is placed out of plane, allowing the cameras to determine the orientation of the vehicle. With only the four in-plane markers in view, a good tracking of position is possible but not a good tracking of attitude. However, when all five markers are in view, both position and attitude can be accurately tracked. The reliability of the position and attitude data is further expounded in the Results section.

The following three experiments were performed in the Vicon^® motion capture system: (1) general flight test in which roll, pitch, yaw, and heave modes were excited, (2) unfolded hand-launched tests while inducing initial roll, pitch, and yaw rates during the launch, and (3) folded hand-launched tests with initial conditions similar to the proof-of-concept testing flight profile shown in Figure 1. These tests sweep all possible launch cases, allowing for a more robust validation of the 6-DOF flight dynamics model.

3.3. Propeller Testing

Propeller thrust and torque were measured at different rotational speeds to provide the input data for the flight dynamics model. The motor utilized on the quadcopter is the Diatone Industries Mamba 1103, 8500 Kv 5 g brushless motor, and it is combined with a durable HQ 65 mm two-bladed propeller. A Nano17 6-axis load cell is utilized to measure propeller thrust and torque using a hover-stand instrumented with a laser tachometer to measure the rotational speed (shown in Figure 9). The 3D-printed post between Nano17 and the motor is three rotor radii in height to ensure that the ground effect does not influence the measurements. Tests were conducted to map the propeller thrust and torque as a function of the RPM in hover.

4. Flight Dynamics Modeling

In order to systematically analyze the performance of the hand-launched MAV, a 6-DOF flight dynamics model was developed. This virtual representation accurately captures the vehicle’s intricate dynamics, encompassing both its translational and rotational motions in three-dimensional space. The model aims to replicate the complex dynamics observed during the launch phase with the propeller arms folded, the transition to stable hover via the propeller arms unfolding, and the actual stable flight by integrating the dynamics and aerodynamics models with an active control system architecture. The developed simulation model can predict the MAV’s behavior from the moment it departs the user’s hand until it lands. It provides information regarding the vehicles’s states, including position, velocity, acceleration, angles, angular rates, and angular acceleration as well as the control inputs during the course of flight. Furthermore, the model also provides the time history of the thrust, torque, and rotational speed of each propeller, offering a comprehensive understanding of the MAV’s performance throughout its flight envelope.

The dynamics model serves a crucial objective, which is to gain insight into the range of initial conditions at which the MAV can be safely launched. For example, it allows one to determine the maximum acceptable tumbling rate (pitch/roll/yaw rate) on release or the minimum apex height from which the MAV could recover. These performance parameters are particularly significant considering two key factors, which are (1) compactness constraints imposed by this design, which limits propeller and motor size, and therefore, the maximum thrust and control authority, and (2) the low inertia of the current vehicle. Obtaining performance limits through flight testing alone is not practical, underscoring the need for a high-fidelity simulation model (or digital twin) of the hand-launched MAV. Leveraging such a simulation model will not only provide a deeper understanding of the vehicle’s capabilities but also pave the way to optimize its parameters to enhance performance and identify the best operating conditions.

This section discusses the development of the governing equations of motion of the hand-launched MAV, the numerical solution procedure, the underlying aerodynamic model, and the design of the control architecture that is integral to the flight dynamic simulations.

4.1. Equations of Motion

Before deriving the equations of motion of the MAV (essentially a quadcopter), it is important to establish the coordinate systems. Reference frames serve as a means to describe the position and orientation of objects in a three-dimensional space. Two pivotal coordinate frames commonly used in rigid body dynamics are the inertial frame and the body frame.

A flat and non-rotating Earth is considered as the inertial frame of reference, denoted $\mathcal{I}$, with its origin located at the home position. Using the North-East-Down (NED) coordinate system, a standard choice in the aeronautical literature [20,21], the inertial frame aligns with ${\widehat{\mathit{x}}}^{\mathcal{I}}$ (x-axis) pointing northward, ${\widehat{\mathit{y}}}^{\mathcal{I}}$ (y-axis) eastward, and ${\widehat{\mathit{z}}}^{\mathcal{I}}$ (z-axis) downward, perpendicular to the X-Y plane, as shown in Figure 10. The body frame, denoted as $\mathcal{B}$, represents a coordinate system rigidly attached to the aircraft. In this frame, the unit vector ${\widehat{\mathit{x}}}^{\mathcal{B}}$ (x-axis) is directed towards the nose of the aircraft, the unit vector ${\widehat{\mathit{y}}}^{\mathcal{B}}$ (y-axis) is positioned to the right of the x-axis from the pilot’s perspective, and the unit vector ${\widehat{\mathit{z}}}^{\mathcal{B}}$ (z-axis) points downwards through the bottom of the aircraft, perpendicular to the X-Y plane. The origin of this coordinate system is located at the center of gravity (CG) of the vehicle. Note that when the orientation of the aircraft changes, the body frame rotates correspondingly at the same angular velocity.

To describe the orientation of the body frame coordinate system with respect to the inertial frame, quaternion (Euler parameters) representation of attitudes is utilized. Quaternions are popular attitude representations that offer a redundant, non-singular attitude description that is well suited to describe arbitrary, large rotations. Given the arbitrary orientation in which the MAV will be thrown, quaternions are the appropriate choice for accurate attitude representation. According to Euler’s theorem, a coordinate system can be brought from an arbitrary initial orientation to an arbitrary final orientation by a single rotation through a principal angle $\mathbf{\Phi }$ about a principal axis $\widehat{\mathit{e}}$, where the principal axis is an axis fixed in both the initial and final orientation. The quaternions $\widehat{\overline{\mathit{q}}}$ are written as a combination of a scalar quantity representing the principal angle of rotation and a vector quantity representing the principal axis of rotation.

$$\widehat{\overline{\mathit{q}}}=\left[\begin{array}{c}q\\ \mathit{q}\end{array}\right]=\left[\begin{array}{c}cos\left({\displaystyle \frac{\mathbf{\Phi }}{2}}\right)\\ sin\left({\displaystyle \frac{\mathbf{\Phi }}{2}}\right)\widehat{\mathit{e}}\end{array}\right]=\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]$$

(1)

The quaternions have to follow the following constraint:

$$∥\widehat{\overline{\mathit{q}}}∥=\sqrt{{\left(q_0\right)}^2+{\left(q_1\right)}^2+{\left(q_2\right)}^2+{\left(q_3\right)}^2}=1$$

(2)

Now, the transformation from the inertial frame $\mathcal{I}$ to the body frame $\mathcal{B}$ in terms of quaternions is given by:

$$R_{\mathcal{I}}^{\mathcal{B}}=I-2q[\mathit{q}\times ]+2{[\mathit{q}\times ]}^2$$

(3)

where $[\mathit{q}\times ]=\left[\begin{array}{ccc}0& -q_3& q_2\\ q_3& 0& -q_1\\ -q_2& q_1& 0\end{array}\right]$ and I is 3 × 3 identity matrix. More details on quaternions can be found at [22].

The quadcopter MAV is treated as a rigid body for deriving the equations of motion [21]. Several state variables are defined to represent the position, orientation, linear, and angular velocities of the quadcopter MAV.

$$\begin{array}{ccc}(x, y, z)\hfill & =\hfill & \mathrm{inertial}\mathrm{position}\mathrm{of}\mathrm{the}\mathrm{aircraft}\hfill \\ (u, v, w)\hfill & =\hfill & \mathrm{body}\mathrm{frame}\mathrm{velocity}\mathrm{of}\mathrm{the}\mathrm{aircraft}\hfill \\ (p, q, r)\hfill & =\hfill & \mathrm{body}\mathrm{frame}\mathrm{angular}\mathrm{velocity}\mathrm{of}\mathrm{the}\hfill \\ \hfill & \hfill & \mathrm{aircraft}\hfill \\ (q_0, q_1, q_2, q_3)\hfill & =\hfill & \mathrm{quaternions}\hfill \end{array}$$

The position of the MAV $(x, y, z)$ is specified in the inertial frame coordinate system $\mathcal{I}$, while the velocity $(u, v, w)$ and angular velocity $(p, q, r)$ are given in the body frame coordinate system $\mathcal{B}$. It is also important to note that the quaternions $(q_0, q_1, q_2, q_3)$ must be related to the angular velocity $(p, q, r)$, which is defined in the body frame coordinate system $\mathcal{B}$. Finally, Newton’s and Euler’s laws are utilized to derive the translational and rotational equations of motion of the vehicle, respectively. The resulting 6-DOF model of the MAV’s kinematics and dynamics can be written as:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]=R_{\mathcal{B}}^{\mathcal{I}}={R_{\mathcal{I}}^{\mathcal{B}}}^T\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$

(4)

$$\left[\begin{array}{c}\dot{u}\\ \dot{v}\\ \dot{w}\end{array}\right]=\left[\begin{array}{c}rv-qw\\ pw-ru\\ qu-pv\end{array}\right]+{\displaystyle \frac{1}{m}}\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]$$

(5)

$$\left[\begin{array}{c}\dot{q_0}\\ \dot{q_1}\\ \dot{q_2}\\ \dot{q_3}\end{array}\right]={\displaystyle \frac{1}{2}}\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]\otimes \left[\begin{array}{c}0\\ p\\ q\\ r\end{array}\right]$$

(6)

$$\left[\begin{array}{c}\dot{p}\\ \dot{q}\\ \dot{r}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \frac{J_y-J_z}{J_x}}qr\\ {\displaystyle \frac{J_z-J_x}{J_y}}pr\\ {\displaystyle \frac{J_x-J_y}{J_z}}pq\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{J_x}}{\tau }_x\\ {\displaystyle \frac{1}{J_x}}{\tau }_y\\ {\displaystyle \frac{1}{J_x}}{\tau }_z\end{array}\right]$$

(7)

Here, ${R_{\mathcal{I}}^{\mathcal{B}}}^T$ represents the transpose of ${R_{\mathcal{I}}^{\mathcal{B}}}^T$, m represents the mass of the MAV, and ⊗ represents quaternion multiplication. As the equations are developed in the body fixed frame, the inertia tensor is diagonal and has diagonal elements $(J_x, J_y, J_z)$. Vectors $(f_x, f_y, f_z)$ and $({\tau }_x, {\tau }_y, {\tau }_z)$ are the net forces and net moments acting on the MAV, respectively, and are expressed in the body frame coordinate system $\mathcal{B}$. The forces and moments acting on the MAV are due to gravity and aerodynamic forces. Since the origin of the body frame is located at the CG of the MAV, gravity does not produce any moments on the aircraft. The force due to gravity expressed in the body frame is written as:

$$f_g=R_{\mathcal{I}}^{\mathcal{B}}\left[\begin{array}{c}0\\ 0\\ mg\end{array}\right]$$

(8)

where m is the mass of the MAV and g is the gravitational constant.

4.2. Aerodynamic Modeling

The propeller is assumed to produce a force along the axis of rotation and a torque about the axis. For the current work, the other aerodynamic forces and moments are assumed to be negligible because the MAV operates almost exclusively in conditions near hover, where translational velocities are extremely small. Under these low advance-ratio conditions, the contributions of translational drag, inflow asymmetry, and other secondary aerodynamic effects are minimal compared to the dominant propeller thrust and torque terms.The thrust T and torque Q generated by the propellers are assumed to scale proportionally to the square of the rotational speed. This can be mathematically described as:

$$T=K_T{\mathrm{\Omega }}^2$$

(9)

$$Q=K_Q{\mathrm{\Omega }}^2$$

(10)

where $K_T$ and $K_Q$ are the thrust and torque constants, respectively. The values are determined through rotor hover testing, with detailed explanations provided in subsequent sections.

The quadcopter MAV is assumed to have a configuration in the body frame as shown in Figure 11. The quadcopter MAV model has a symmetric structure with respect to the geometric center O of the vehicle. Each rotor is located at $L_1$ and $L_2$ distances along the x-axis and y-axis, respectively, from the center. It is important to note that the geometric center O is not the origin of the body frame; instead, the CG of the aircraft is designated as the origin of the body frame. The geometric center is located at the coordinates ($x_0$, $y_0$) relative to the CG of the aircraft. Rotors 2 and 4 produce a negative pitching moment, while rotors 1 and 3 produce a positive pitching moment. Similarly, rotors 1 and 2 generate a negative rolling moment, whereas rotors 3 and 4 produce a positive rolling moment. Finally, rotors 1 and 4 create an anti-clockwise (-ve) yawing moment while rotors 2 and 3 create a clockwise (+ve) yawing moment.

The total forces and moments from all four rotors acting on the MAV in the body frame can be expressed as:

$$f_{aero}=\left[\begin{array}{c}0\\ 0\\ K_T({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_3^2+{\mathrm{\Omega }}_4^2)\end{array}\right]$$

(11)

$${\tau }_{aero}=\left[\begin{array}{cccc}-K_T(L_1-x_o)& -K_T(L_1+x_o)& K_T(L_1-x_o)& K_T(L_1+x_o)\\ K_T(L_2+x_o)& -K_T(L_2+x_o)& K_T(L_2-x_o)& -K_T(L_2-x_o)\\ -K_Q& K_Q& K_Q& -K_Q\end{array}\right]\left[\begin{array}{c}{\mathrm{\Omega }}_1^2\\ {\mathrm{\Omega }}_2^2\\ {\mathrm{\Omega }}_3^2\\ {\mathrm{\Omega }}_4^2\end{array}\right]$$

(12)

Now, combining aerodynamic force and gravity, the net forces and moments acting on the MAV can be expressed in the body frame as follows:

$$f_{net}=\left[\begin{array}{c}{f}_x\\ f_y\\ f_z\end{array}\right]=f_{aero}+f_g$$

(13)

$${\tau }_{net}=\left[\begin{array}{c}{\tau }_x\\ {\tau }_y\\ {\tau }_z\end{array}\right]={\tau }_{aero}$$

(14)

4.3. Controller Architecture

Coupling a controller model to the dynamic model is essential for achieving stable flight and also for capturing the transient dynamics of the vehicle. Once the folded MAV is thrown in the air, the control system is only activated after the propeller arms unfold, which is similar to the strategy employed on the physical flying prototype. The initial objective of the flight controller on the actual prototype is to rapidly stabilize the MAV from a tumbling state to a trim state before responding to any pilot commands. In the simulated environment, the controller emulates this behavior by prioritizing the stabilization of the MAV upon activation. Once stabilized, it seamlessly transitions to tracking the designated trajectories to accomplish specific flight objectives.

A conventional quadcopter control strategy [21] is adopted to determine control actions (rotational speed of the propellers). A quadcopter can move directly along the body-fixed vertical axis (z-axis) without affecting any other attitude as the thrust generated by each rotor is aligned along the vertical direction. The vehicle can heave up or down by changing the RPM of all rotors equally. However, to maneuver along the horizontal x- and y-axes and alter the heading, the attitude must be adjusted. Consequently, four independent controllers have been developed to stabilize the system around the desired positions in x, y, and z, as well as the desired heading. As a result, the following control inputs are required:

• ${\mathrm{\Omega }}_{mean}$: the desired increase in RPM across all motors that facilitates vertical motion along the z-axis.
• ${\mathrm{\Omega }}_{{\Delta }_x}$: the desired difference in RPM between the motors on either side of the x-axis leads to changes in the roll angle, resulting in lateral motion along the y-direction.
• ${\mathrm{\Omega }}_{{\Delta }_y}$: the desired difference in RPM between the motors on either side of the y-axis leads to changes in the pitch angle, resulting in longitudinal motion along the x-direction.
• ${\mathrm{\Omega }}_{{\Delta }_z}$: the desired difference in RPM between the clockwise and counterclockwise motors generates a net torque that rotates the vehicle around the vertical z-axis.

The controller estimates these four control inputs, which results in a desired change in RPM for each propeller.

4.3.1. Vertical Motion Controller (Z-Controller)

The motion along the vertical z-axis is controlled using two cascaded PID loops, as depicted in Figure 12. The first PID loop controls the position, while the second PID loop regulates the velocity. The desired altitude of the vehicle is obtained from the desired trajectory. The position error along the z-axis is the input to the first PID loop, which regulates the desired velocity along the z-axis. The output of this PID loop is then used as input for the second PID loop, which calculates the velocity error along the z-axis. The final output of the controller is the desired mean RPM (${\mathrm{\Omega }}_{mean}$) for each motor. Adjusting the RPMs of each motor by an equal amount of (${\mathrm{\Omega }}_{mean}$), the thrust on each rotor is affected equally, allowing the aircraft to move vertically up or down. This approach ensures that changing the RPMs does not affect any other control parameters, allowing smooth and stable vertical motion control.

4.3.2. Lateral Motion Controller (X/Y-Controller)

This controls the motion along the horizontal x- and y-axes through the use of four cascaded PID loops for position, velocity, angle, and angular rates, as illustrated in Figure 13. The first two layers of the controller are the position and velocity PID loops, which output the desired velocities and angles about the x(roll)- or y(pitch)-axis, respectively. These two PID controllers are followed by an angle and angular-rate PID controller, which takes the error between the desired angles/desired angular rate and the current angles/current angular rate of the vehicle as input. The angle and angular rate PID together ensure that the pitch and roll angles remain steady, without abrupt changes. The output of the angular rate PID loop, which is the final output of this controller is the difference in RPM between the motors on either side of the y-axis (left and right motors) leading to changes in the roll angle (${\mathrm{\Omega }}_{{\Delta }_x}$) or the difference in RPM between the motors on either side of the x-axis (front and rear motors) that leads to changes in the pitch angle (${\mathrm{\Omega }}_{{\Delta }_y}$).

However, unlike the vertical motion controller, the RPMs of each motor cannot be changed equally. For example, to achieve forward pitch motion, the RPMs of the front motors should be decreased, and the RPMs of the back motors should be increased to create a non-zero net moment in the forward direction. The (${\mathrm{\Omega }}_{{\Delta }_x}$) for this case is distributed among the motors in such a way that there is a net forward moment without affecting net thrust, net sideward moment, and net yawing moment. The same applies to a sideward roll motion, where the RPMs of the left and right motors are adjusted to create a net moment in the desired direction.

4.3.3. Heading Controller (Yaw-Controller)

The heading controller is used to control the aircraft heading and is operated by controlling both the heading angle and the yaw rate. It does this using two cascaded PID loops, namely, an angle and an angular rate PID, to achieve the desired heading and yawing rate specified by the user, as shown in Figure 14. The angle PID takes the difference between the desired heading angle and the current heading angle as input and outputs the desired angular velocity/rate about the vertical z-axis. This angular velocity/rate is then fed into the angular rate PID, which takes the difference between the desired angular velocity and the current angular velocity as input. The final output of the heading controller is the desired difference in RPMs between the clockwise and counterclockwise rotors, denoted as (${\mathrm{\Omega }}_{\Delta z}$). This generates a moment that rotates the vehicle around the vertical z-axis. For example, if the desired objective is to rotate the MAV clockwise, all clockwise motors are uniformly increased by (${\mathrm{\Omega }}_{{\Delta }_z}$) from the nominal value, while all counterclockwise motors are uniformly decreased by the same amount. This adjustment ensures that the net thrust and net moments about the x- and y-axes remain unchanged, while producing a positive clockwise moment in the aircraft.

Table 3. PID controller gains used in simulation.

Control Loop	Axis	P	I	D
Position	X	3.0	0.0	0.0
	Y	3.0	0.0	0.0
	Z	4.0	0.0	0.0
Velocity	VX	30.0	0.0	0.0
	VY	30.0	0.0	0.0
	VZ	10,000.0	3000.0	0.0
Attitude	Roll	15.0	15.0	0.0
	Pitch	15.0	15.0	0.0
	Yaw	15.0	15.0	1.0
Attitude Rate	Roll	100.0	100.0	0.0
	Pitch	200.0	200.0	0.0
	Yaw	500.0	500.0	0.0

summarizes the PID gains used for each control axis and loop level in the simulation environment. It can be noted that the attitude PID gains in simulation differ from those used in real flight. This mismatch can be attributed to the fact that in simulation, position commands generate smooth attitude setpoints, whereas in real flights, the pilot inputs attitudes directly. Furthermore, the simulated mapping from attitude rates to rotor RPM differs from the actual hardware, resulting in a different effective rate response and consequently different gains in the simulation.

5. Results

This section discusses the results of flight testing as well as the 6-DOF flight dynamics model. Several experiments were conducted to validate the ability of the flight controller to stabilize the MAV from a tumbling state, thus validating the proof of concept. A motion capture system was utilized to obtain accurate trajectory data of the aircraft post-deployment, aiding in the development and validation of the simulation model. To obtain an accurate aerodynamic model of the propeller for the simulation, isolated propeller tests were performed using a 6-axis load cell, providing thrust and torque measurements at different rotational speeds. Upon completion of all these tests, the necessary parameters for the simulation were obtained and, finally, the flight dynamic model was validated through several flight experiments conducted at the Vicon^® facility.

5.1. Proof-of-Concept Testing

Since these tests were not conducted at the Vicon^® facility, only the attitude data was obtained from the flight controller. The difference between the commanded and the actual roll, pitch, and yaw rates (rate attitude error) could quantify the performance of the control system. When the error is zero, it means that the aircraft is perfectly following the commanded input. The roll and pitch attitude rate errors during the 2 s of the flight test from the start of the throw to stable hover are shown in Figure 15. The different phases of the flight profile, shown in Figure 1, are also labeled on the plots. The yaw rate response did not vary much in this proof-of-concept test and, therefore, it is omitted here. If the MAV is just hovering stably, the errors will be close to zero, as can be seen towards the end of the flight. As demonstrated in Figure 15, the ability of the current MAV to recover from such high tumbling rates (2500 deg/s) to a stable hover within a fraction of a second further illustrates the true potential of this vehicle for the rapid/robust deployment scenarios mentioned in the Introduction section.

5.2. Flight Testing in Motion Capture System

It is imperative to validate the accuracy of the Euler angle, quaternion, and position data obtained through the Vicon^® motion capture system before utilizing it to validate the 6-DOF model. The attitude data are validated utilizing the results from a general flight test (not hand-launched). Figure 16 compares the time history of the roll, pitch, and yaw angle data estimated by the Vicon^® system compared to the one estimated based on the IMU data from the flight controller. Although Vicon^® data are noisier than flight controller data, it is clear that the general trends of the data are similar. Specifically, the Root Mean Square Error (RMSE), Equation (15), was computed between the IMU and Vicon^® attitude data, where the roll RMSE, pitch RMSE, and yaw RMSE are 6.9°, 5.2°, and 1.9°, respectively. Despite the RMSE being relatively low, the IMU data from the flight controller are utilized to validate the attitude estimates of the 6-DOF model as it has less noise. One of the main reasons for the noise in attitude is due to the small size of the vehicle forcing the markers to be close to each other, which is not ideal for the estimation of attitude from the position data of the marker. Noise could be reduced with additional out-of-plane reflective markers; however, only one marker was utilized to preserve the mass and inertia of the original vehicle.

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n}{(y_i-{\widehat{y}}_i)}^2}{n}}}$$

(15)

On the other hand, the Vicon^® system was very successful in estimating the position data. Although there is no position data available from the flight controller, the motion capture position data can be qualitatively validated by comparing images of the tracked flight trajectory and the actual flight trajectory. A stacked image of the MAV’s trajectory from a video of the actual launch is shown in Figure 17a. Similarly, a stacked image from the screen capture of the MAV as the Vicon^® object in the software is shown in Figure 17b alongside the position data obtained shown in Figure 17c as a three-dimensional trajectory. In comparison to the two stacked images and the three-dimensional position data, the trajectories qualitatively appear to be the same.

5.3. Propeller Testing

The thrust and torque of the propeller were measured over a range of rotational speeds using the hover test stand. The experimental measurements were then used to develop a best-fit relationship between thrust and RPM, as well as torque and RPM, for implementation in the vehicle dynamics model. The thrust and torque results are shown in Figure 18. The best-fit line was found using Equations (9) and (10), which are applicable under hover conditions. Given that the rotors operate at high rotational speeds and the MAV remains close to hover throughout its operation, the resulting advance ratios are extremely small, which makes this assumption valid. From the best fit, $K_T$ and $K_Q$ are found to be $6.27\times {10}^{-10}$ and $3.459\times {10}^{-12}$, respectively. The maximum thrust measured for a single motor-propeller is approximately 0.55 N. As a result, the rotational speed required to hover the 112 g (1.10 N) MAV is approximately 21,000 RPM, as each propeller needs to produce around 0.275 N thrust.

5.4. Flight Dynamics Model Validation

All of the above experiments contributed to the determination of the crucial parameters necessary to establish an accurate simulation model. Now, real flight test data can be compared with simulation model predictions to assess its accuracy. One notable point to consider is that, as previously mentioned, the position data from the motion capture system are accurate while the attitude data are noisy compared to the data obtained from the flight controller’s IMU. Hence, the experimentally obtained attitude data from the flight controller’s IMU and position data from the Vicon^® motion capture system are utilized to validate the 6-DOF model. The position data from Vicon^® are also used to calculate the inertial velocities, and the flight controller’s gyroscope can provide the attitude rate information. Three different flight test cases are compared with the simulation: (1) general MAV flight, (2) unfolded hand-launch with imposed initial attitude rates, and (3) folded hand-launch with imposed initial attitude rates.

5.4.1. Case 1: General MAV Flight

The flight dynamics model is first compared for the simplest case: a general flight test around hover where the roll, pitch, yaw, and heave modes were excited. The primary aim of simulating this scenario is to calibrate the controller in the simulation model to match that of the actual MAV. The physical MAV controller operates using a set of cascaded PIDs and the inputs to these PIDs are scaled versions of angles and angular rates derived from the stick motions provided by the pilot. Without knowledge of this mapping and the scaling factors involved, replicating the exact controller setup in simulation would be challenging. Hence, a conventional quadcopter control approach is adopted, where the position data from the Vicon^® system serve as the input to the controller in the simulation. The outer position and velocity PIDs are configured to mimic the pilot’s commands, while the inner attitude and attitude rate controller PIDs emulate those of the onboard flight controller. The model also takes in initial position, velocity, attitude, and angular rates as inputs and a full 70 s flight was simulated. The resulting position is then compared against the motion capture data in Figure 19. As expected, there is minimal difference in the simulated and experimental positions because they are direct inputs into the model. The velocity is then provided as an output from the position PID controller; this simulated velocity is compared to the derived velocity from the motion capture experiments in Figure 20. Velocity is not a direct input into the model and therefore, some variation between the simulated and experimental is expected, but there is still minimal error for this case. The velocity PID controller then outputs a commanded pitch and roll attitude, while yaw is a direct input into the heading controller. The simulation results are then compared against the flight controller IMU data in Figure 21. The simulation results show a good correlation with the experimental roll and pitch attitude even though it is another step away from the input data. The yaw prediction is almost an exact fit on the experimental data because it is also a direct input to the PIDs. The attitude PID controller then outputs the angular rates and they are compared with the gyro data in Figure 22. Despite the angular rates being three control loops away from the inputs of the x- and y-positions, the simulation still agrees well with the experimental data. Finally, the RPM commanded by the 6-DOF model is compared with the true RPM of each motor from the flight controller in Figure 23. The motor RPM is shown starting after one second to better display the relevant data because the commanded RPM is minimal during the first second. There is a significant mismatch at the beginning of the flight in the RPM data. This can be attributed to the ground effect since this was not modeled in the simulation. It is evident in Figure 19, that the MAV was flying at a very low altitude in the beginning. It is worth noting that the simulated motors can respond instantaneously, which is a fair assumption given their small sizes and the physical motor at this scale can reach the commanded RPM in a few milliseconds. Hence, it can be assumed to be instantaneous. Apart from the initial 10–15 s, the simulated RPM values correlate closely with the experimental ones outside of noise. The quantitative agreement between simulation and experiments is summarized using the RMSE metric described in Equation (15). For this test case, the RMSE values were found to be 0.15 m for position, 0.22 m/s for velocity, 2.5° for attitude, 9.9 deg/s for angular rate, and 1037 RPM for motor speed.

5.4.2. Case 2: Unfolded Hand-Launch

The next flight case brings an increase in complexity. The model must now start by simulating the free motion of the MAV under the influence of only gravity with the initial conditions provided during launch (free fall) until the motors are switched on. As soon as the motors are on, the objective of the flight controller is to stabilize the MAV in the shortest time possible. The pilot starts giving commands at some time instant after the stabilization, and the vehicle follows the command. The simulation model also starts with the simulation of the free fall region and then prioritizes stabilizing the MAV before executing a trajectory tracking control. The free fall phase typically lasts for less than 0.5 s depending on how fast the pilot issues stabilization commands to the MAV. Despite its brief duration, this is a crucial region and dictates the state of the MAV just before the motors start rotating. It is also important to input an accurate set of initial conditions into the simulation because at the moment the vehicle leaves the user’s hand, it could have high linear and angular velocities. Obtaining these values from flight data can be challenging and can have a huge impact on the final state of the MAV due to its low inertia. After free fall, when the pilot activates the stabilization mode, the flight controller autonomously stabilizes the vehicle regardless of any other pilot commands. This region of stabilization also only lasts for less than a second, and after that it responds to the pilot’s commands. As mentioned earlier, the simulated controller is not exactly the same as the actual flight controller on the physical MAV, but rather a conventional controller that is tuned to behave like the actual controller. Hence, most of the discrepancies in the simulation results compared to the experimental data can be attributed to inaccurate initial conditions and small differences in the flight controller. Furthermore, the motion capture data and flight controller IMU data are collected with different start times, leading to minor inconsistencies when mapping the two instances to the same timescale.

Despite these complexities, the 6-DOF model was able to fairly accurately reproduce the true flight dynamics of the MAV. Similar to case 1, the simulated position has minimal deviation from the experimentally measured position and can be seen in Figure 24. The velocity comparison, shown in Figure 25 introduces more inconsistencies as it moves farther away from the input of position, but still shows good agreement between the model and experimental velocity. Attitude estimates from the simulation are imperfect at the beginning due to the aforementioned complexities, but still follow similar trends, and quickly after the pilot starts controlling the MAV shows minimal disagreements with the experimental results. The comparison of attitude with the flight controller IMU data is shown in Figure 26. The angular rates have the highest deviation during launch and also take longer to stabilize (shown in Figure 27); however, they still follow a trend similar to the experimental data. Finally, the motor RPM comparisons, Figure 28, show disagreement at the beginning but followed them well toward the end. The simulation in case 2 shows more deviation from the test data than in case 1, but this is expected due to the added complexity of the launch sequence. The level of agreement between the simulated and experimental data for this case is quantified using the RMSE calculation method described in Equation (15). The calculated RMSE values are 0.16 m for position, 0.34 m/s for velocity, 9° for attitude, 58.6 deg/s for angular rate, and 2421 RPM for motor speed. Despite some inaccuracies, the 6-DOF model mimics the true flight dynamics fairly accurately and helped to further tune the model before simulating the most complex scenario: Case 3.

5.4.3. Case 3: Folded Hand-Launch

Case 3 adds further complexity because of the need to model the unfolding of the propeller arms in flight during the hand-launch. The unfolding takes about 0.05 s. A key change that is happening to the MAV during the unfolding process is in its moments of inertia, and this will affect its rotational dynamics. The principal moments of inertia of the MAV are calculated from its detailed CAD model, and it was observed that the moments of inertia of the MAV increase by a factor of three from the folded ($J_x=3\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$, $J_y=8\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$, $J_z=11\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$) to unfolded ($J_x=9\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$, $J_y=23\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$, $J_z=31\times {10}^{-5} \mathrm{kg}·{\mathrm{m}}^2$) configuration. To account for this time variation in rotational inertia in flight, the simulation model linearly interpolates the principal inertia with time during the unfolding phase. This is an approximation and might introduce some minor differences in the dynamics. Additionally, any error in determining the precise time instance when unfolding begins for both the motion capture data and the IMU data could further contribute to the discrepancies. The hand-launch during case 3 was more aggressive than case 2 and the imposed initial rates caused the MAV to flip three times before stabilizing. Case 3 also encompasses all of the complexities mentioned in both cases 1 and 2. Once again, the simulated position has minimal deviation from the experimental motion capture data and can be seen in Figure 29. The velocity comparison is also fairly accurate, as shown in Figure 30. The attitude and angular rate comparisons, shown in Figure 31 and Figure 32, respectively, showcase some disagreement during the unfolding region. This is due to the difficulty in estimating the rotor unfolding and motor spin-up times. Finally, the motor RPM can be seen in Figure 33, and while the simulated RPMs have a greater disagreement with the true motor RPM at the beginning, it still provides an accurate estimation of the true motor RPM throughout the flight profile. The RMSE values for this case are as follows: position 0.15 m, velocity 0.4 m/s, attitude 20.7°, angular rate 76.5 deg/s, and motor speed 1535 RPM. Despite the added complexities of a hand-launch that causes the MAV to flip three times before stabilization, as well as the unfolding of the propeller arms, the 6-DOF model replicates the actual flight dynamics with a high degree of accuracy. As a result, the model is successfully validated and can be utilized in future studies to estimate the initial conditions under which the MAV can be launched and stabilized successfully.

6. Discussion

The main goal of the present work has been to demonstrate the feasibility of a hand-launched foldable MAV that is small enough to fit inside a shirt pocket. The MAV features a custom-built main body with rotatable propeller arms and a maximum dimension of only 11 cm when folded. This extreme portability makes it well suited for rapid deployment in the field, particularly in scenarios where conventional storage and deployment methods for UAVs are impractical. Once deployed, the arms unfold in flight to assume a quadcopter configuration. A control methodology has been implemented to autonomously recover the MAV to a stable hover from a hand launch, even with arbitrary initial velocity and angular rates. Several outdoor flights and indoor flight experiments are conducted in a motion capture facility to demonstrate the feasibility of the design. These experiments demonstrated reliable performance across a range of launch conditions and confirmed the MAV’s ability to stabilize in less than two seconds after being hand launched. Finally, a comprehensive 6-DOF flight dynamics model of the MAV is developed and validated using the flight test data. This model serves as a valuable tool to analyze limitations and improve the overall optimality of the design. The key findings of this study are listed below.

• The locking mechanism designed for the propeller arms exhibited quick unfolding of the propeller arms while in flight. It takes less than 0.05 s to fully unfold the propeller arms. This deployment mechanism can be highly advantageous in emergencies as it does not rely on any standard launch procedure; instead, a simple hand launch suffices.
• The proficiency of the control architecture was demonstrated through numerous indoor and outdoor flight tests. Flight data was collected from the motion capture system and on-board sensors on the flight controller. The analysis reveals that the MAV was able to recover in less than a second even when launched with high linear and angular velocities.
• A 6-DOF simulation model of the quadcopter MAV accurately predicted various aspects of the MAV’s state, including the control inputs. Three distinct flight scenarios, including a general MAV flight, an unfolded throw, and a full hand-launch from a folded position, are simulated and validated. The accuracy of the prediction is heavily dependent on the initial conditions, as evidenced by the results of the general MAV flight scenario, where acquiring initial conditions is relatively straightforward. Despite significant changes in inertia mid-flight due to the deployment of the arms, the simulation predictions remain largely unaffected due to the rapid deployment of the arms and immediate spinning up of the motors.

The results of this work open up the opportunity for a number of applications where portability and rapid deployment are highly valued. Potential use cases include search and rescue missions in constrained environments, quick response inspection of hazardous areas, and field reconnaissance in remote or inaccessible regions. Building on the current platform, future work will be focused on developing the autonomy of the foldable MAV platform. This includes the integration of onboard perception and navigation systems towards achieving complete autonomous launch, target detection, and tracking capability in GPS-denied or cluttered environments. Advanced guidance and control algorithms will be developed for the MAV to enable following moving targets or performing waypoint navigation with minimal human intervention. In addition, onboard sensing and real-time decision-making advances will be explored to improve robustness against environmental disturbances such as wind and turbulence. These advances will expand the flight envelope of the MAV and render it feasible for a more extensive range of applications, such as search and rescue, surveillance, and environmental monitoring.

7. Patents

U.S. nonprovisional patent application No. 19/221,846 filed 29 May 2025, and entitled “Foldable Micro Air Vehicles”.