VR-Supported Analysis of UAV—Magnetic Launcher’s Cart System

Abstract

The subject of the research is a model of a magnetic launcher, which is an innovative alternative to commercially occurring unmanned aircraft launchers (UAV). As the take-off is an energy-demanding phase of the flight; therefore, abandoning the power supply of the UAV during this phase significantly affects increasing the potential range and duration of UAV flight. The magnetic launcher offers the significant advantage of minimizing friction between the starting cart and the launcher, resulting in the higher energy efficiency of the system. Research conducted so far has shown that the possibility of accelerating the aircraft on the longer runway offered by the launcher reduces aircraft overloads occurring during take-off. As a result, the launcher, aircraft, and onboard equipment are much safer. This paper presents the system’s mathematical modeling and numerical simulation results for micro-class UAV take-off and landing using the analyzed magnetic launcher. The computer program for analyzing system dynamics was implemented in the MATLAB environment. Simulation results were visualized graphically and as animations in Virtual Reality. The VR application was implemented in Unity and ran on VR goggles Oculus Quest2. The simulations carried out show that—in the absence of control—an important factor reducing the takeoff distance and affecting the aircraft load is the adoption of a non-zero takeoff thrust of the UAV. The initial pitch angle also has a significant impact on the takeoff process. With an increase in this parameter, the length of the takeoff distance decreases and the lift-off speed decreases, but too much pitch angle may result in the aircraft descending in the first moments of flight, which could lead to a collision with the launch rails.

1. Introduction

Electric propulsion is currently a widely used propulsion system for small and medium-sized UAVs. However, this results in a smaller permissible weight of the carried equipment, a shorter range, a lower altitude, and a shorter flight time. The most energy-demanding phase of the flight is take-off, therefore—as demonstrated by the studies on magnetic and electromagnetic suspensions described below—the use of a magnetic launcher during take-off and landing, assisted by an electric linear motor, will positively impact the operational efficiency of the UAV.

The phenomenon of levitation has been stimulating the imagination and inspiring scientists for a long time. The technology that is most often referred to in the context of levitation involves suspensions based on magnetic and electromagnetic interactions. Already at the beginning of the 20th century, designers of future transport systems have explored the use of magnetic levitation to eliminate friction between wheels and ground. Herman Kemper, a German engineer, was one of the pioneers in this domain. The studies on the phenomenon of levitation he started in 1922 led to the creation of Transrapid—a high-speed monorail using electromagnetic suspension (EMS) in 1966. Transrapid lines were tested independently in Germany and Japan, and the first 31 km long commercial line linking the Shanghai Airport with downtown Pudong was established in 2004 in China [1].

Research involving the technology of magnetic levitation has been uninterruptedly continued until today. An alluring alternative to systems employing electromagnetic suspension is the high-temperature superconducting (HTS) maglev developed in China [2], Brazil (MagLev²-Cobra) [3], or Germany (the SupraTrans system) [4,5]. Using high-temperature superconductors in transport system suspensions entails numerous advantages. The levitation mechanism using superconductors does not require heavy electromagnets to be installed in the vehicle, making HTS technology lighter and simpler compared to EMS [3]. Superconductors also enable stable levitation, even for weak magnetic fields. Due to magnetic flux trapping, a superconductor maintains the shape of the field, where it was introduced into the superconducting state and always tends to the initial position [6], which enables not only superconductor levitating above fixed magnets but also maintaining stable levitation in any other superconductor position relative to the magnet (it can be suspended under the magnet or to the side).

As evident, superconductor levitation is justified and promising. It arouses growing interest among researchers, primarily owing to the potential technical applications. A brief review of studies on available technologies seems to confirm the thesis that superconductor levitation technology is a promising implementation in terms of the analyzed operation of unmanned aerial vehicle magnetic launcher that enables collision-free take-off and landing.

Take-off and landing are some of the most important maneuvers executed by any aircraft, and the details of these procedures depend on UAV class and type. The take-off of most unmanned aerial vehicles requires using a launcher or aviation catapults. Rocket systems, as well as rubber, pneumatic and hydraulic launchers, are the means currently used the most [7,8]. Typically, for objects weighing up to 30 kg, simple catapults are used, in which the propulsion system is composed of a rope with high elastic properties, and the rope tension is managed to control the speed of the UAV. Although these launchers allow for controlling the UAV speed by changing the tension of the flexible elements, the acceleration at the moment of releasing the tension increases rapidly, reaching values exceeding 200 m/s² [7,8]. For heavier UAVs weighing up to 100 kg, pneumatic or hydraulic launchers are used. In this type of catapult, the accelerations increase slightly more slowly, and the maximum acceleration values usually do not exceed 50–100 m/s². The rate of increase in overloads can be controlled to some extent [9,10]. If the launch weight of the UAV exceeds 150 kg, launch rockets are used. Heavier UAVs often take off from airports, just like conventional planes. Electromagnetic launchers provide precise control over acceleration. There are no restrictions on the weight of the aircraft with electromagnetic catapults (they are used on modern aircraft carriers [11]), and the increase in acceleration can be precisely controlled [12,13].

Catapults that utilize magnetic forces are an interesting alternative. There has recently been noticeable progress in the work on such systems, addressed mainly to the needs and interests of the military. Navies of global leaders strive to retrofit their aircraft carriers with aircraft take-off and landing systems based on magnetic levitation. The leading example, in this case, is EMALS (Electromagnetic Aircraft Launching System) and the AAG (Advanced Arresting Gear) braking system developed by General Atomics for the US Navy [12,13]. The developed technology demonstrated that, compared to conventional steam devices, electromagnetic catapults accelerate aircraft in a smoother manner, reducing the load on their airframes; they are also smaller, generate less noise and their operation and maintenance are easier. Unfortunately, the launcher’s application scope is restricted by the levitation force, which appears only when the velocity of a taking of aircraft is higher than 5 m/s.

This disadvantage is not the case with a launcher that employs passive suspension with superconductors [14]. Its technological prototype was developed under the GABRIEL (Integrated Ground and on-Board system for Support of the Aircraft safe Take-off and Landing) project implemented as part of the seventh framework program of the European Union [15,16]. One of the objectives of the GABRIEL project was to demonstrate the feasibility of its assumptions through the development of a test bench. The test bench was designed to showcase the high-temperature superconductors used in the levitation suspension system of the launch cart [17]. However, one of the significant shortcomings of the GABRIEL system validation studies was the requirement of a flying vehicle that can reach speeds of about 2 m/s, as the test bench’s takeoff and landing magnetic path was relatively short. Due to budget constraints, the research stand was limited to a magnetic track length of approximately 6 m. Therefore, a tilt-wing micro-class aircraft, designed by Aachen University of Technology, was chosen as the GABRIEL test vehicle mainly because of its structural simplicity compared to other V/STOL (Vertical/Short Takeoff and Landing) designs.

The aircraft takeoff and landing tests were successful, which demonstrated the feasibility and effectiveness of the solution [18]. However, a number of issues were observed in the course of the research. These included the lack of propulsion calibration allowing the aircraft to achieve lift-off velocity over a given track length; selecting take-off parameters such as the pitch angle of the aircraft relative to the launch cart; difficulties in mastering the landing procedure, allowing for the safe placement of the UAV on the launch cart. These issues revealed a need to analyze the launcher system dynamics, which was a motivation behind this paper.

An important element of the conducted analyses was the use of virtual reality (VR) to depict the behavior of the element of the modeled magnetic launcher system over time. Therefore, the results of conducted UAV launcher take-off and landing process numerical simulations were implemented in VR software. The main advantages of this solution include the ability to directly observe aircraft and cart position and orientation at take-off and landing, changes of the scene observation perspective introduced by the user, as well as stopping, rewinding and slowing down simulation time during key movements of system elements.

VR (Virtual Reality) technology is increasingly making its way into the world of science and industry, where it is used in areas such as product design and modeling, product servicing, and employee training. The enormous potential of VR lies in its ability to engage all the senses of the user, while also providing time and cost savings. Therefore, VR is already widely used as an educational tool during aircraft crew training [19] and airport personnel training [20,21]. On the other hand, VR technology is increasingly being used to create 3D models and visualizations for virtual prototyping. For example, article [22] addressed the issue of remote and immersive control of UAVs based on VR, while [23] undertook the development of bionic sensor systems for visual detection of UAV collisions. This article presents a new way of presenting engineering simulation results using a VR environment. An application was created that allows for 3D visualization of the trajectory and flight parameters of UAVs. It is possible to manipulate the observation perspective of the launcher system, control the simulation time, and display UAV flight parameters including velocity vector and angle of attack from within the program.

The following is the organization of this paper. Section 2 reviews the structure of a passive magnetic launcher analyzed within the paper, describes a launcher dynamics system model and presents proprietary VR visualization software to analyze the results of launcher numerical simulations conducted in MATLAB. Section 3 contains the results of numerical simulations, including three different aircraft take-off scenarios and one landing scenario example. The obtained results have been discussed in Section 4, while Section 5 concentrates on the conclusions and further research prospects.

2. Materials and Methods

2.1. A Magnetic Launcher Prototype

The magnetic launcher, which was developed and constructed as part of the GABRIEL project, consisted of a magnetic suspension system and a linear drive. The launcher was composed of six sections/plates, each of which measured 960 mm in length. Each section was made up of a base measuring 20 × 400 × 960 mm, two rails made from PA9 measuring 10 × 51 × 960 mm, a rail for the stator fastened to the base, a system of connecting rails with the base, and a system of connecting tools (Figure 1a). Each magnetic rail was affixed with three rows of neodymium magnets (Nd2Fe14B), each measuring 15 × 15 × 5 mm. This means that each rail of each test bench section contained 192 magnets. The total length of the test bench was 5760 mm.

The plates were equipped with angle brackets to enable the assembly of the linear drive and supports for the magnetic track assembly and adjustment. The UAV to be launched was placed on a magnetic sledge, which served as a movable element of the launcher. The sledge was constructed using a rigid aluminum frame weighing 0.8 kg and was equipped with magnetic suspension elements as well as a linear electric motor (Figure 1b). A system of guiding pins with special tightening screws, equipped with guiding cones, was used to connect the boards. The test bench’s layout is depicted in Figure 2.

The HIWIN LMC A6 linear motor was used for acceleration. The motor consisted of stators connected in one straight edge (Figure 3a) and a forcer (Figure 3b). The HIWIN LMC A6 linear motor (Figure 1b) was controlled by an inverter that provided a constant frequency and voltage ratio. This was the basic control method. The basic specifications of the HIWIN LMC A6 linear motor are as follows: peak force of 180 N, continuous force of 60 N, force constant of 54.5 N, maximum velocity of 5 m/s, maximum acceleration of 100 m/s², repeatability of +/−0.001 mm, accuracy of +/−0.005 mm.

The forcer of the linear motor was mounted to the sledge frame. The system was designed to allow for adjustment of the forcer’s position relative to the tracks. Additionally, small rolling bearings were placed on the sledge to serve as simple wheels, limiting the forcer’s movement in the event of any side movements caused by transitions between the tracks or unbalance.

The linear motor is powered by a single-phase inverter, with acceleration and deceleration curves programmed into it. The direct frequency change is realized by an external voltage varying in the range. Additionally, the inverter is controlled by three logic signals responsible for the runner’s movement to the right, movement to the left, and the blocking of inverter operation. Unblocking or activating the direction of movement is conducted by giving a low signal level of “0” logical. The inverter can also be manually controlled by setting the frequency using the numeric keypad.

The start and landing aid system is mounted on the magnetic sledge and includes a movable platform that automatically adjusts itself relative to the angular position of the aircraft. The platform and aircraft were developed by a research team from Aachen. Figure 4a shows a sledge with a platform, and Figure 4b shows a VTOLMAV aircraft. The total mass of the sledge, landing system, and aircraft is 8 kg.

For the V/STOL aircraft, a tilt-wing design was chosen, mainly due to its structural simplicity in comparison to other V/STOL designs. With this design, the wing can be tilted between a horizontal position for fast flight and a vertical position for slow flight or even hover. During a very slow flight with the wing in a near-vertical position, the major part of the weight of the aircraft is supported by the thrust of the two engines, which are fixed to the wing and tilted with it, providing an upward thrust vector. One particular challenge in controlling this type of aircraft is the ambiguity of the effects of the control surfaces, which depend on the flight speed and tilt angle of the wing.

Figure 4. (a) GABRIEL MAV levitation landing platform on levitating sledge; (b) VTOL class micro aerial vehicle during landing.

Figure 4. (a) GABRIEL MAV levitation landing platform on levitating sledge; (b) VTOL class micro aerial vehicle during landing.

Magnets neodymium (Nd2Fe14B) included in the magnetic track rail were polarized in the smallest dimension, i.e., height. The magnet columns are arranged in opposite directions, with unipolar magnets along the tracks and polypolar magnets across the tracks (see Figure 5a). This arrangement creates a “chute” magnetic field over the tracks (Figure 5b). Along the track line, the magnetic field gradient is zero, which eliminates the force that would brake the cart’s motion. Across the tracks, the non-zero magnetic field gradient “pushes” the cart towards the center, significantly enhancing suspension stability. The take-off cart hovers above the tracks owing to the levitation force generated between the high-temperature superconductors and the neodymium magnets creating the magnetic field.

Four containers, referred to as “boxes”, were attached to the corners of the take-off cart (Figure 1b). Each container held four YBCO superconductors, which were shaped like cylinders measuring 21 mm in diameter and 8 mm in height and had a critical temperature of 92 K. The boxes, which included the containers with superconductors, had a total mass of 80 g. After filling the containers with liquid nitrogen during the tests, the superconductors transitioned into a state of superconductivity, generating a levitation force between the high-temperature superconductors and the permanent magnets fixed to the launcher tracks, producing the magnetic field. The low thermal conductivity of the container material helped to maintain the superconductors at a temperature below critical for an extended period, which was a crucial requirement for the cart to levitate above the magnets during aircraft take-off or landing.

The take-off procedure involves placing an unmanned aerial vehicle on a take-off cart above the launcher tracks before take-off (Figure 3a). The cart is accelerated using a linear electric motor. Once the take-off velocity is achieved, the UAV lifts off from the cart and transitions to the climb range, while the take-off cart decelerates and either returns to its starting position or prepares to receive a landing UAV.

The procedure is reversed during the touchdown (Figure 6b). The task of the cart is to intercept a landing aircraft. UAV and take-off cart position, orientation, as well as angular and linear velocities, are monitored and synchronized, so as to minimize the reaction force upon UAV-cart collision. The aircraft levels the flight and then touches down on the cart. The system then enters the braking range, the sleigh loses energy and once the aircraft comes to a complete stop, together with the cart moves to the parking position. The system then prepares for the next take-off or landing.

2.2. System Dynamics Model

The UAV launcher that utilizes passive magnetic suspension with superconductors has been modeled as a configuration of three elements. These include a launcher base with tracks fitted with permanent magnets that generate a magnetic field, a take-off cart powered by a linear motor, and hovering over the magnetic tracks through the Meissner effect, and an unmanned aerial vehicle that takes off or lands with the use of the launcher cart (Figure 7).

The system movement constraints arise from the interactions between its individual elements, i.e., magnetic tracks and the launcher cart, and the cart and taking off/landing aircraft. The value and direction of such interactions may change during movement, a detailed analysis of the modeling process covering the forces acting upon the magnetic launcher is discussed in [25].

The developed physical model assumed that the system dynamics are impacted by mechanical actions (values and moments of gravitational, propulsion and reactive forces resulting from the aircraft’s contact with the cart’s surface, as well as aerodynamic force values and moments acting upon the aircraft), and take-off cart magnetic levitation forces associated with the Meissner effect. As demonstrated in [23,26], levitation force values and moments depend on the strength and orientation of the magnetic field generated by launcher tracks, as well as the position and orientation of levitating superconductors. The levitation force resultant acting upon the take-off cart has been modeled as the total value and moment of forces acting upon four boxes of the take-off cart [24]. These actions result from the effect on the superconductors in each of the boxes, with the levitation force showing little attenuation. The extent of vertical vibration attenuation in the cart is proportional to the vertical components of the cart’s velocity and the attenuation factor that has been experimentally determined [18,24].

The variety of force application points and the force and force moment directions depending on the track-cart-UAV configuration enforce high regularity and uniqueness of the system modeling process. For this purpose, each launcher system element should have a separate coordinate system assigned. The diagrams in Figure 7 show the position of individual coordinate system elements: inertial O_fx_fy_fz_f—closely connected with the ground; magnetic O_mx_my_mz_m—used as a reference to describe the magnetic levitation forces; O_sx_sy_sz_s—take-off cart system and O_bx_by_bz_b—aircraft system. A detailed description of the position and orientation of individual coordinate systems can be found in [26].

Rigid body and flight mechanics modeling methods, as well as fundamental laws of electromagnetism, were used to derive a mathematical model for the aircraft-launcher cart-magnetic track system dynamics. The description of system dynamics has been based on the principle of momentum and torsion changeability. The determined equations are of matrix form, which is natural to such computational suites as MATLAB.

The analyzed dynamics model assumed that an unmanned aerial vehicle that is taking-off is rigidly connected with the take-off cart frame until lift-off, thus, the position, as well as the angular and linear velocities of the aircraft were constantly zero relative to the cart. In such a case, the UAV-cart system has 6 degrees of freedom, and its movement within the relative O_sx_sy_sz_s system, related to the take-off cart, is described by a system of matrix Equation (1). In the case of the analyzed notation, vector and tensor superscripts mean a coordinate system wherein a given vector was described, while subscripts indicate the start of a given system and are relative to the given system described. The operator $\mathfrak{s}\left(x\right)$ defined for a 3D vector creates an antisymmetric tensor, which enables writing a vector product using a numerical matrix and vector product that is convenient to implement.

Cart-UAV system dynamics equations for the take-off procedure take the following form:

$$\begin{array}{c}\left[\begin{array}{cc}\left(m_{\left(s\right)}+m_{\left(b\right)}\right)\mathbb{I}& -m_{\left(s\right)}\mathfrak{s}\left(r_{g_s/s}^s\right)-m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/s}^s\right)\\ m_{\left(s\right)}\mathfrak{s}\left(r_{g_s/s}^s\right)+m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/s}^s\right)& {\Im }_{\left(s\right)s}^s+{\Im }_{\left(b\right)s}^s-m_{\left(b\right)}\left(\mathfrak{s}\left(r_{g_b/b}^s\right)\mathfrak{s}\left(r_{b/s}^s\right)+\mathfrak{s}\left(r_{b/s}^s\right)\mathfrak{s}\left(r_{g_b/b}^s\right)\right)\end{array}\right]\left[\begin{array}{c}{a}_{s/f}^{s\left(ff\right)}\\ {\epsilon }_{s/f}^{s\left(f\right)}\end{array}\right]+\\ \left[\begin{array}{c}-\mathfrak{s}\left({\omega }_{s/f}^s\right)\left(m_{\left(s\right)}\mathfrak{s}\left(r_{g_s/s}^s\right)+m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/s}^s\right)\right){\omega }_{s/f}^s\\ \mathfrak{s}\left({\omega }_{s/f}^s\right)\left({\Im }_{\left(s\right)s}^s+{\Im }_{\left(b\right)s}^s-m_{\left(b\right)}\left(\mathfrak{s}\left(r_{g_b/b}^s\right)\mathfrak{s}\left(r_{b/s}^s\right)+\mathfrak{s}\left(r_{b/s}^s\right)\mathfrak{s}\left(r_{g_b/b}^s\right)\right)\right){\omega }_{s/f}^s\end{array}\right]=\left[\begin{array}{c}{F}_{\left(s\right)}^s+F_{\left(b\right)}^s\\ M_{\left(s\right)s}^s+M_{\left(b\right)b}^s+\mathfrak{s}\left(r_{b/s}^s\right)F_{\left(b\right)}^s\end{array}\right]\end{array}$$

(1)

where:

• $\mathbb{I}$—3 × 3 identity matrix,
• $m_{\left(s\right)}$, $m_{\left(b\right)}$—take-off cart and unmanned aerial vehicle mass, respectively,
• ${\Im }_{\left(s\right)s}^s$, ${\Im }_{\left(b\right)s}^s$—take-off cart and UAV inertia tensor in the O_sx_sy_sz_s system,
• $\mathfrak{s}\left(r_{g_s/s}^s\right)$, $\mathfrak{s}\left(r_{g_b/s}^s\right)$—take-off cart and UAV center of mass displacement vector antisymmetric tensor in the O_sx_sy_sz_s system,
• $a_{s/f}^{s\left(ff\right)}$—absolute linear acceleration of the cart,
• ${\epsilon }_{s/f}^{s\left(f\right)}$—absolute angular acceleration of the cart,
• ${\omega }_{s/f}^s$—absolute angular velocity of the cart,
• $F_{\left(s\right)}^s$, $F_{\left(b\right)}^s$—forces acting upon the cart and UAV, analyzed in the O_sx_sy_sz_s system,
• $M_{\left(s\right)s}^s$, $M_{\left(b\right)s}^s$—moments of forces acting upon the cart and UAV.

An element coupling all parts of the magnetic launcher system is the take-off cart. Its movement results from gravitational and aerodynamic actions, linear motor propulsion and magnetic levitation forces. Therefore, the resultant value $F_{\left(s\right)}^s$ and moment of the forces $M_{\left(s\right)}^s$ acting upon the take-off cart, and taken into account in the motion equations, are described by the following relationships:

$$F_{\left(s\right)}^s=F_{\left(Gs\right)}^s+F_{\left(As\right)}^s+F_{\left(Ls\right)}^s+F_{\left(Ns\right)}^s,$$

(2)

$$M_{\left(s\right)s}^s=M_{\left(Gs\right)s}^s+M_{\left(As\right)s}^s+M_{\left(Ls\right)s}^s,$$

(3)

where:

• $F_{\left(Gs\right)}^s$, $M_{\left(Gs\right)s}^s$—value and moment of gravitational forces acting upon the cart;
• $F_{\left(As\right)}^s$, $M_{\left(As\right)s}^s$—resultant value and moment of aerodynamic forces acting upon the cart;
• $F_{\left(Ls\right)}^s$, $M_{\left(Ls\right)s}^s$—resultant value and moment of magnetic levitation force;
• $F_{\left(Ns\right)}^s$—cart propulsion force.

The motion of an unmanned aerial vehicle during take-off is impacted by the gravitational force, aerodynamic actions and UAV propulsion force. Therefore, the resultant value $F_{\left(b\right)}^s$ and moment of the forces $M_{\left(b\right)s}^s$ acting upon the take-off cart, and taken into account in the motion equations, are described by the following relationships:

$$F_{\left(b\right)}^s=F_{\left(Gb\right)}^s+F_{\left(Ab\right)}^s+F_{\left(Nb\right)}^s,$$

(4)

$$M_{\left(b\right)}^s=M_{\left(Gb\right)b}^s+M_{\left(Ab\right)b}^s,$$

(5)

where:

• $F_{\left(Gb\right)}^b$, $M_{\left(Gb\right)b}^b$—value and moment of gravitational forces acting on UAV;
• $F_{\left(Ab\right)}^b$, $M_{\left(Ab\right)b}^b$—resultant value and moment of aerodynamic forces;
• $F_{\left(Nb\right)}^b$—UAV propulsion force.

Upon the landing of the aircraft on the take-off cart, the force of the UAV impacting the cart surface is modeled as the ground restoring force. As a result, it was assumed that after the aircraft touches the take-off cart, system elements are impacted by values ${\overrightarrow{F}}_{\left(r\right)}$ and moments ${\overrightarrow{M}}_{\left(r\right)r}$ of reactive forces, which were modeled using an elastic-attenuated element. The model considers the relative motion of the aircraft after landing, resulting in a UAV-cart system with twelve degrees of freedom. The O_rx_ry_rz_r system (Figure 7) starts at the moment the aircraft touches the take-off cart surface for the first time. The distance between the cart and the reactive system changes due to platform deformations caused by the landing aircraft. Reactive force values and moments are proportional to the change in distance between the cart’s center of mass and the reactive system initiation point, as well as the aircraft’s relative velocity ${\overrightarrow{v}}_{b/s}^{\left(s\right)}$ and angular velocity ${\overrightarrow{\omega }}_{b/s}$ within the cart-related system.

In the case in question, the dynamic equations for the cart-UAV system during landing take the following form:

$$\begin{array}{c}\left[\begin{array}{cccc}{m}_{\left(s\right)}{\mathbb{I}}_{3x3}& 0& -m_{\left(s\right)}\mathfrak{s}\left(r_{\frac{g_s}{s}}^s\right)& 0\\ m_{\left(b\right)}{\mathbb{I}}_{3x3}& m_{\left(b\right)}{\mathbb{I}}_{3x3}& -m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{g_b}{b}}^s\right)-m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{b}{s}}^s\right)& -m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{g_b}{b}}^s\right)\\ m_{\left(s\right)}\mathfrak{s}\left(r_{\frac{g_s}{s}}^s\right)& 0& {\Im }_{\left(s\right)s}^s& 0\\ m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{g_b}{b}}^s\right)& m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{g_b}{b}}^s\right)& {\Im }_{\left(b\right)b}^s-m_{\left(b\right)}\mathfrak{s}\left(r_{\frac{g_b}{b}}^s\right)\mathfrak{s}\left(r_{\frac{b}{s}}^s\right)& {\Im }_{\left(b\right)b}^s\end{array}\right]\left[\begin{array}{c}{a}_{s/f}^{s\left(ff\right)}\\ a_{b/s}^{s\left(ss\right)}\\ {\epsilon }_{s/f}^{s\left(f\right)}\\ {\epsilon }_{b/s}^{s\left(s\right)}\end{array}\right]\hfill \\ +\left[ \begin{array}{c}-m_{\left(s\right)}\mathfrak{s}\left({\omega }_{s/f}^s\right)\mathfrak{s}\left(r_{g_s/s}^s\right){\omega }_{s/f}^s\\ -m_{\left(b\right)}\mathfrak{s}\left({\omega }_{b/f}^s\right)\mathfrak{s}\left(r_{g_b/b}^s\right){\omega }_{b/f}^s-m_{\left(b\right)}\mathfrak{s}\left({\omega }_{s/f}^s\right)\mathfrak{s}\left(r_{b/s}^s\right){\omega }_{s/f}^s-m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/b}^s\right)\mathfrak{s}\left({\omega }_{s/f}^s\right){\omega }_{b/s}^s\\ \mathfrak{s}\left({\omega }_{s/f}^s\right){\Im }_{\left(s\right)s}^s{\omega }_{s/f}^s\\ \mathfrak{s}\left({\omega }_{b/f}^s\right){\Im }_{\left(b\right)b}^s{\omega }_{b/f}^s-m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/b}^s\right)\mathfrak{s}\left({\omega }_{s/f}^s\right)\mathfrak{s}\left(r_{b/s}^s\right){\omega }_{s/f}^s+{\Im }_{\left(b\right)b}^s\mathfrak{s}\left({\omega }_{s/f}^s\right){\omega }_{b/s}^s\end{array} \right]\hfill \\ +\left[\begin{array}{c}0\\ 2m_{\left(b\right)}\mathfrak{s}\left({\omega }_{s/f}^s\right)v_{b/s}^{s\left(s\right)}\\ 0\\ 2m_{\left(b\right)}\mathfrak{s}\left(r_{g_b/b}^s\right)\mathfrak{s}\left({\omega }_{s/f}^s\right)v_{b/s}^{s\left(s\right)}\end{array}\right]=\left[\begin{array}{c}{F}_{\left(s\right)}^s\\ F_{\left(b\right)}^s\\ M_{\left(s\right)s}^s\\ M_{\left(b\right)b}^s\end{array}\right]+\left[\begin{array}{c}{F}_{\left(r\right)}^s\\ -F_{\left(r\right)}^s\\ M_{\left(r\right)r}^s+\mathfrak{s}\left(r_{r/s}^s\right)F_{\left(r\right)}^s\\ -M_{\left(r\right)r}^s-\mathfrak{s}\left(r_{r/s}^s\right)F_{\left(r\right)}^s\end{array}\right]\hfill \end{array}$$

(6)

where:

• $F_{\left(r\right)}^s$, $M_{\left(r\right)r}^s$—value and moment of the reactive force between the cart and UAV;
• $\mathfrak{s}\left(r_{r/s}^s\right)$—antisymmetric tensor of the vector associated with the displacement of the O_rx_ry_rz reactive system starts relative to the O_sx_sy_sz_s system.

The resulting dynamic motion equations were supplemented with kinematic and geometric relationships determining the behavior of system elements in the course of the UAV taking-off from the launcher. These are:

• Absolute angular velocity of the take-off cart:

$${\omega }_{s/f}^s={\displaystyle \int }{R}_f^s{\epsilon }_{s/f}^{f\left(f\right)}dt={\displaystyle \int }{\epsilon }_{s/f}^{s\left(f\right)}dt,$$

(7)

Absolute angular velocity of the UAV:

$${\omega }_{b/f}^s={\displaystyle \int }{R}_s^b{\epsilon }_{s/f}^{s\left(f\right)}dt,$$

(8)

Absolute linear velocity of the take-off cart:

$$v_{s/f}^{s\left(f\right)}={\displaystyle \int }\left(a_{s/f}^{s\left(ff\right)}-\mathfrak{s}\left({\omega }_{s/f}^s\right)v_{s/f}^{s\left(f\right)}\right)dt,$$

(9)

Absolute linear velocity of the UAV:

$$v_{b/f}^{b\left(f\right)}={\displaystyle \int }\left(a_{b/f}^{b\left(ff\right)}-\mathfrak{s}\left({\omega }_{b/f}^b\right)v_{b/f}^{b\left(f\right)}\right)dt,$$

(10)

Shift speed of quasi-Eulerian angles determining the cart’s spatial orientation:

$$\left[\begin{array}{c}{\dot{\varphi }}_{s/f}\\ {\dot{\theta }}_{s/f}\\ {\dot{\psi }}_{s/f}\end{array}\right]=\left[\begin{array}{ccc}1& \sin \left({\varphi }_{s/f}\right)\tan \left({\theta }_{s/f}\right)& \cos \left({\varphi }_{s/f}\right)\tan \left({\theta }_{s/f}\right)\\ 0& \cos \left({\varphi }_{s/f}\right)& -\sin \left({\varphi }_{s/f}\right)\\ 0& \sin \left({\varphi }_{s/f}\right)\cos \left({\theta }_{s/f}\right)& \cos \left({\varphi }_{s/f}\right)/\cos \left({\theta }_{s/f}\right)\end{array}\right]{\omega }_{s/f}^f,$$

(11)

Shift speed of quasi-Eulerian angles determining the UAV’s spatial orientation:

$$\left[\begin{array}{c}{\dot{\varphi }}_{b/f}\\ {\dot{\theta }}_{b/f}\\ {\dot{\psi }}_{b/f}\end{array}\right]=\left[\begin{array}{ccc}1& \sin \left({\varphi }_{b/f}\right)\tan \left({\theta }_{b/f}\right)& \cos \left({\varphi }_{b/f}\right)\tan \left({\theta }_{b/f}\right)\\ 0& \cos \left({\varphi }_{b/f}\right)& -\sin \left({\varphi }_{b/f}\right)\\ 0& \sin \left({\varphi }_{b/f}\right)\cos \left({\theta }_{b/f}\right)& \cos \left({\varphi }_{b/f}\right)/\cos \left({\theta }_{b/f}\right)\end{array}\right]{\omega }_{b/f}^f,$$

(12)

Take-off cart position relative to the inertial system:

$$r_{s/f}^f={\displaystyle \int }\left(R_s^f{v}_{s/f}^{s\left(f\right)}\right)dt,$$

(13)

UAV position relative to the inertial system:

$$r_{b/f}^f={\displaystyle \int }{R}_b^f{v}_{b/f}^{b\left(f\right)}dt,$$

(14)

where:

• $R_f^s$, $R_s^b$—rotation matrices describing the orientation of the inertial system relative to the take-off cart or the cart system relative to the aircraft system, respectively,
• ${\epsilon }_{s/f}^{s\left(f\right)}$—absolute angular acceleration of the cart,
• $a_{s/f}^{s\left(ff\right)}$—absolute linear acceleration of the cart,
• $v_{s/f}^{s\left(f\right)}$—absolute linear velocity of the cart,
• ${\omega }_{s/f}^s$—absolute angular velocity of the cart,
• $r_{s/f}^f$—vector of the distance between the cart’s local system and the inertial system.

The derived mathematical system of the magnetic launcher constitutes grounds for further research involving the dynamic properties of its motion. The next section discusses the numerical simulation process of an unmanned aerial vehicle taking-off and landing using a magnetic launcher take-off cart.

2.3. VR Application Usage

The study involved using proprietary VR visualization software to analyze the results of MATLAB numerical simulations of the UAV take-off and landing process with a magnetic launcher take-off cart. The software program was implemented in the Unity environment [27]—one of the top engines for developing computer games and other interactive materials. The process of implementing interactive functions associated with VR, such as GUI via controllers, was facilitated owing to the use of ready-made XR Interaction Toolkit library components. To enhance VR software development, utilizing a Software Development Kit (SDK) can prove to be an invaluable tool. SDKs offer a pre-built and pre-configured set of interactions that can be easily incorporated into VR projects. Among the free SDKs available, the XR Interaction Toolkit stands out as one of the most popular and is the one used in our project. The toolkit is equipped with script libraries that allow for the implementation of various VR interactions, such as object manipulation, user interface interaction, hand gesture recognition, locomotion systems, and physical interactions. With the help of an SDK such as the XR Interaction Toolkit, VR developers can accelerate their projects by leveraging pre-built tools and libraries, ultimately leading to more efficient development and better VR experiences for users.

The program we have implemented enables the execution of simulations on the Oculus Quest 2, a VR headset created by Oculus, which is a division of Meta. The Quest 2 is a standalone device that operates wirelessly, running games and software on an Android-based operating system. It is capable of six degrees of freedom positional tracking, allowing for a more immersive VR experience. The Quest 2 includes a pair of high-resolution VR goggles, as well as two symmetric controllers that enable user hand tracking. These controllers enable users to interact with the virtual environment by pointing at objects with a ray or by pressing the controller buttons. With its advanced hardware capabilities, the Oculus Quest 2 is an ideal platform for running simulations, and our program takes full advantage of the device’s features to provide an engaging and realistic VR experience.

In the Unity scene depicted in Figure 8, users are presented with a detailed model of a sample MAV called the E-flite Opterra [28]. The scene also includes models of the launcher’s cart and magnetic tracks, as well as carefully crafted lighting to enhance the immersive experience. A camera is included that tracks the user’s head movements, further increasing the realism of the simulation. To enable users to interact with certain elements of the scene, a Graphical User Interface (GUI) is provided. Additionally, the scene includes markings that show the distance traveled by the cart, providing users with helpful visual feedback. Overall, the scene is designed to provide a rich and engaging VR experience that fully leverages the capabilities of the Unity engine and the Oculus Quest 2 hardware.

To enable users to control the UAV simulation in our program, the Oculus Quest 2 controllers are utilized. Users can manipulate the controllers to increase or decrease the simulation’s speed, as well as to pause or rewind the simulation. The trigger button, as shown in Figure 9, is used to stop or rewind the simulation, with the magnitude of the rewind being proportional to the pressure applied to the button. The user can also decelerate the simulation by pressing a push button, with the rate of deceleration depending on the force applied. Additionally, haptic feedback in the form of controller vibrations may occur at specific moments during the simulation, such as when the UAV lands on the cart. These features allow users to fully immerse themselves in the simulation, providing a more realistic and engaging experience.

One of the most significant advantages of visualizing UAV and cart motion in VR is the user’s ability to observe it from any distance or perspective. This allows the user to freely change the camera position and move around the virtual scene, providing a more immersive experience. To switch to a different pre-defined camera position, the user can simply press one of the controller buttons X/Y/A/B, as illustrated in Figure 10. The program includes five sets of camera positions, each offering a unique perspective, allowing the user to view the simulation from a variety of angles. This feature further enhances the user’s ability to explore and interact with the virtual environment, providing a more engaging and realistic experience:

• When no button is pressed, the program defaults to a stationary camera positioned at eye level and oriented horizontally. This provides users with a stable point of view from which they can observe the simulation and serves as a useful starting point for exploring the virtual environment. This camera is set perpendicularly to the tracks in the place of the cart starting position, in the origin of the O_fx_fy_fz_f coordinate system.
• B button pressed—a moving camera perpendicular to the tracks and tracking UAV motion and O_bx_by_bz_b coordinate system.
• A button pressed—a moving camera perpendicular to the tracks and tracking cart motion and O_sx_sy_sz_s coordinate system.
• Y button pressed—a stationary camera parallel to the tracks, set at the starting position of the cart.
• X button pressed—a moving camera positioned above the nose of the UAV and parallel to the tracks.

Figure 9. Application operation via VR controllers.

Figure 9. Application operation via VR controllers.

Figure 10. Camera positions corresponding to X/Y/A/B buttons on VR controllers.

Figure 10. Camera positions corresponding to X/Y/A/B buttons on VR controllers.

Regardless of the selected camera position, if a user turns his/her head to the left, the user will see a user interface (Figure 11) that can be used to restart the simulations and select a simulation result file generated in MATLAB. The GUI also allows the user to turn on/off the visualization of selected motion parameters, such as the UAV’s velocity vector and the vector pointing longitudinal axis of the UAV. The angle between those vectors determines the angle of attack.

The program features a panel that is attached to the user’s right hand, as depicted in Figure 11b. This panel displays the current values of selected parameters, including time, UAV height, UAV velocity, pitch angle, and attack angle. In addition, it includes plots generated in MATLAB that show the UAV and cart’s spatial trajectory, as well as the height, velocity, pitch angle, and attack angle of the UAV and cart. These features provide users with a wealth of information about the simulation and allow them to track key metrics in real-time, enhancing their ability to understand and interact with the virtual environment.

3. Results

The implemented numerical model of the take-off and landing process utilized the presented matrix form of motion Equations (1) and (6), supplemented with (2)–(5) and (7)–(14) relations. The levitation force value was approximated based on experimental tests and MES analysis discussed in [18,24,25]. It was assumed that the motion of the system is considered under calm weather conditions, with the dynamic properties of the launch cart and micro-UAV being dependent on the environment through changes in pressure and density of the surrounding air. It was assumed that the base of the launcher is stationary and located at sea level, where the air density is 1.225 kg/m³. The mass and dimensions of the platform were neglected in the simulations. The mass of the launch cart was assumed to be 1.4049 kg. It was assumed that the position of the center of mass of the unloaded cart remains constant, and its inertia tensor was described by Equation (15).

$${\Im }_s=\left[\begin{array}{ccc}0.02& 0& 0\\ 0& 0.0364& 0\\ 0& 0& 0.1491\end{array}\right] {\mathrm{kgm}}^2,$$

(15)

The linear motor driving the starting cart is modeled using a constant thrust force in the direction of the cart’s motion. The mass and geometric parameters of the electric-powered UAV were also assumed as constant values. The aim of the conducted numerical simulations was to analyze the take-off and landing process of a fixed-wing UAV configuration, which could not be accomplished within the scope of the GABRIEL project. Therefore, a micro-class UAV with a weight of 1.5 kg, a reference area of 0.2961 m², and a wing span of 0.84 m was taken into account as a test UAV (Figure 12). During flight, the displacement of the UAV’s center of mass is 0.088125 m, the average aerodynamic chord is 0.3525 m, and the inertia tensor is described by Equation (16).

$${\Im }_b=\left[\begin{array}{ccc}0.0184& 0& -0.0002159\\ 0& 0.0367& 0\\ -0.0002159& 0& 0.055\end{array}\right] {\mathrm{kgm}}^2,$$

(16)

The simulation model developed in MATLAB was formulated as a starting point for ordinary differential equations. Numerical integration algorithms based on the finite difference method, Runge–Kutta method of the 4th order—“ode45” function or the Gear method, and the “ode15s” function in the case of rigid (ill-conditioned) problems were used to calculate the model.

The visualization and interaction with the VR environment are innovative elements of the developed numerical simulations. This enables direct observation of selected UAV and cart parameters at take-off and landing, the user changing the scene observation perspective, as well as accelerating, slowing down, or stopping the simulation at any time, to thoroughly analyze and interpret an interesting simulation piece. Such a combination of UAV and cart movement parameter graphs generated in MATLAB with a VR animation showing their relative position and orientation provides a full insight into the dynamics of an analyzed magnetic launcher system.

3.1. Simulation of the UAV-Cart System at Take-Off

According to the take-off procedure described in Section 2.1, numerical testing was conducted under the assumption that the UAV at the initial moment was located on the take-off cart above the launcher tracks. It was assumed that at t = 0, the UAV was set on a take-off platform (Figure 4b) fixed {r_b/s}_zs = 0.1 [m] above the take-off cart, while the center of the system related to the aircraft was exactly above the take-off cart system {r_b/s}_xs = 0 m.

Three examples of a UAV take-off from a take-off cart surface were compared to analyze system parameters that enable effective UAV take-off from the analyzed magnetic launcher. All simulations assumed that the take-off cart was driven by a force of 60 N until the UAV lifted off. After the aircraft takes off from the cart, it decelerates.

Simulation 1.

UAV propulsion off, ${\theta }_{b/s}=0 rad$.

The first simulation assumed that a taking-off UAV had its propulsion deactivated and a zero pitch angle relative to the launcher (${\theta }_{b/s}=0 \mathrm{rad}$). The conducted simulations indicated that the aircraft lifted-off from launcher track carts at $t=2.7 \mathrm{s}$ and at a velocity of 58.5 m/s, while the runway was 85 m in this case.

Simulation 2.

UAV propulsion on, ${\theta }_{b/s}=0.03 rad$.

The second simulation assumed a UAV taking-off with an active propulsion of 3 N and a non-zero pitch angle relative to the launcher (${\theta }_{b/s}=0.03 \mathrm{rad}$). As indicated by the simulation, it significantly shortens the runway and reduced lift-off velocity. The UAV lifts-off from the launcher cart at $t=1.1 \mathrm{s}$, at a velocity of 28 m/s and after traveling 21 m.

Simulation 3.

UAV propulsion on, ${\theta }_{b/s}=0.043 rad$.

The third simulation also assumed UAV taking-off with active propulsion (3 N); however, this time with a slightly greater initial pitch angle of ${\theta }_{b/s}=0.043 \mathrm{rad}$. In this case, the UAV detaches from a launcher cart even faster ($t=0.95 \mathrm{s}$), at a lower velocity (24 m/s), and after covering 14.5 m.

Numerical simulation results, generated using MATLAB, which depict the values of selected parameters for the UAV and take-off cart, are presented in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. Based on the data obtained regarding the position and orientation of individual launcher system elements, a VR visualization was created to illustrate the interaction between the cart and a taking-off UAV. Figure 16, Figure 17 and Figure 18 show images from a VR animation that depict selected UAV take-off moments, which are important from the perspective of system element behavior, for the three analyzed simulation cases. A quantitative representation of these changes is provided in Table 1, Table 2 and Table 3.

UAV and cart trajectories at take-off are shown in Figure 13. A navy-blue line marks the cart’s center of mass movement, and a purple line marks the position of the launcher tracks. The green and red waveforms visible to the left and right reflect the position of the cart boxes. The starting position of the local aircraft system is marked with a black line, while the blue line describes the position of aircraft wing tips.

A change in UAV altitude relative to the take-off cart is shown in Figure 14. The trajectory shape is associated with the initial conditions and depends on UAV velocity at lift-off and its pitch angle relative to the cart.

Figure 14. UAV and take-off cart altitude within a relative system, linked to the cart.

Figure 14. UAV and take-off cart altitude within a relative system, linked to the cart.

Figure 15 shows UAV velocities for three analyzed simulations. Lift-off in the first simulation occurs at a velocity of 58.5 m/s, after which an unpropelled aircraft loses its speed. In the case of the other two simulations, when it was assumed that the thrust of a taking-off UAV was non-zero, the lift-off occurred faster and at half the velocity, which impacted airframe load.

Figure 15. UAV velocity.

Figure 15. UAV velocity.

Figure 16 and Figure 17 show take-off cart and UAV pitch angle waveforms. The pitch angle of the take-off cart in all simulations takes negative values at first, and then increases over time and stabilizes around zero after UAV lift-off. In addition, it can be noted that the maximum take-off cart pitch is the lower, the lower the UAV pitch angle during take-off. It is important for operating reasons, since an excessive cart pitch may lead to take-off cart boxes hitting the tracks and damaging it.

Figure 16. Take-off cart pitch angle.

Figure 16. Take-off cart pitch angle.

Figure 17. UAV pitch angle.

Figure 17. UAV pitch angle.

Figure 18 shows the UAV angle of attack waveforms in the analyzed simulations. The vibrations of the angle of attack at take-off result from the tilting motion of the take-off cart. Angles of attack at which a UAV takes-off are increasing in successive simulations. A greater angle of attack implies a higher lift force and faster aircraft lift-off. The angle of attack of aircraft independent flight stabilizes at approx. 0.04 rad. The angle of attack upon take-off in the last simulation is greater than the angle of attack for equilibrium conditions. Reducing the angle of attack during the first moments of independent flight leads to reducing the lift force, hence, loss of UAV altitude (Figure 14). An altitude loss is compensated by increasing UAV velocity; however, this may result in the aircraft colliding with the take-off cart or tracks.

Figure 18. UAV angle of attack.

Figure 18. UAV angle of attack.

The motion of the UAV-cart system during take-off has been visualized in a VR environment, where Figure 19, Figure 20 and Figure 21 display selected images depicting individual seconds of the motion. These seconds are most important due to the behavior of the launcher system elements on the runway. In the background of the launcher, posts indicating successive runway length in meters have been placed. The velocity vector of the UAV, represented by a green arrowhead, shows the direction, sense, and value of the velocity at a given moment. The inclination angle of the velocity vector relative to the brown arrow, which is an extension of the airframe longitudinal axis, determines the UAV angle of attack at that moment in time. A quantitative summary of the main motion parameters for each simulation case has been included in Table 1, Table 2 and Table 3.

Figure 19. Take-off VR visualization images, simulation 1.

Figure 19. Take-off VR visualization images, simulation 1.

Table 1. The basic flight parameters corresponding to the data shown in Figure 19 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.0441	0.0044	12.1460	0.0004
1.0	12.0982	0.0049	24.0420	0.0002
2.0	47.5198	0.0053	46.4800	0.0026
2.5	73.3234	0.0063	56.6600	0.0052
2.6	79.0863	0.0069	57.6130	0.0064
2.7	84.9216	0.0901	57.5958	0.0825
2.8	90.7088	0.5985	56.6019	0.1664
2.9	96.3392	1.5342	55.5602	0.2425

Table 1. The basic flight parameters corresponding to the data shown in Figure 19 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.0441	0.0044	12.1460	0.0004
1.0	12.0982	0.0049	24.0420	0.0002
2.0	47.5198	0.0053	46.4800	0.0026
2.5	73.3234	0.0063	56.6600	0.0052
2.6	79.0863	0.0069	57.6130	0.0064
2.7	84.9216	0.0901	57.5958	0.0825
2.8	90.7088	0.5985	56.6019	0.1664
2.9	96.3392	1.5342	55.5602	0.2425

Table 2. The basic flight parameters corresponding to the data shown in Figure 20 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.1963	0.0048	12.7530	0.0296
0.8	8.1518	0.0054	20.2910	0.0298
1.0	12.7043	0.0058	25.2490	0.0302
1.1	15.3508	0.0061	27.7030	0.0304
1.2	18.1586	0.0114	28.1220	0.0526
1.3	20.9710	0.0520	28.1300	0.0581
1.4	23.7830	0.1223	28.1265	0.0700
1.5	26.5932	0.2239	28.1126	0.0818
1.6	29.4002	0.3571	28.0884	0.0926
1.7	32.2025	0.5211	28.0526	0.1037
1.8	34.9989	0.7157	28.0078	0.1147
2.0	40.5684	1.1948	27.8870	0.1363

Table 2. The basic flight parameters corresponding to the data shown in Figure 20 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.1963	0.0048	12.7530	0.0296
0.8	8.1518	0.0054	20.2910	0.0298
1.0	12.7043	0.0058	25.2490	0.0302
1.1	15.3508	0.0061	27.7030	0.0304
1.2	18.1586	0.0114	28.1220	0.0526
1.3	20.9710	0.0520	28.1300	0.0581
1.4	23.7830	0.1223	28.1265	0.0700
1.5	26.5932	0.2239	28.1126	0.0818
1.6	29.4002	0.3571	28.0884	0.0926
1.7	32.2025	0.5211	28.0526	0.1037
1.8	34.9989	0.7157	28.0078	0.1147
2.0	40.5684	1.1948	27.8870	0.1363

Figure 20. Take-off VR visualization images, simulation 2.

Figure 20. Take-off VR visualization images, simulation 2.

Table 3. The basic flight parameters corresponding to the data shown in Figure 21 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.1962	0.0047	12.7530	0.0396
0.9	10.3031	0.0058	22.7740	0.0398
1.0	12.6673	0.0062	24.0560	0.0397
1.1	15.0768	0.0042	24.1350	0.0381
1.2	17.4942	−0.0005	24.2141	0.0373
1.3	19.9195	−0.0072	24.2941	0.0366
1.4	22.3527	−0.0152	24.3721	0.0362
1.5	24.7938	−0.0241	24.4502	0.0360
1.6	27.2425	−0.0330	24.5272	0.0361
1.7	29.6989	−0.0415	24.6021	0.0364
1.8	32.1629	−0.0489	24.6761	0.0370
2.0	37.1126	−0.0581	24.8190	0.0388

Table 3. The basic flight parameters corresponding to the data shown in Figure 21 (the moment of UAV launch from the launcher is highlighted in bold).

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	3.1962	0.0047	12.7530	0.0396
0.9	10.3031	0.0058	22.7740	0.0398
1.0	12.6673	0.0062	24.0560	0.0397
1.1	15.0768	0.0042	24.1350	0.0381
1.2	17.4942	−0.0005	24.2141	0.0373
1.3	19.9195	−0.0072	24.2941	0.0366
1.4	22.3527	−0.0152	24.3721	0.0362
1.5	24.7938	−0.0241	24.4502	0.0360
1.6	27.2425	−0.0330	24.5272	0.0361
1.7	29.6989	−0.0415	24.6021	0.0364
1.8	32.1629	−0.0489	24.6761	0.0370
2.0	37.1126	−0.0581	24.8190	0.0388

Figure 21. Take-off VR visualization images, simulation 3.

Figure 21. Take-off VR visualization images, simulation 3.

3.2. Simulation of the UAV-Cart System at Landing

The developed magnetic launcher system model and simulation software also enable analyzing the process of an aircraft landing on the launcher cart; however, the developed software does not have implemented algorithms for guiding the aircraft onto the cart and controlling the velocity or orientation of the take-off cart. This issue goes beyond the objective of this study.

The analyzed simulation example assumed that the UAV approaches landing from an altitude of 20 m, at a velocity of 21 m/s and with an engine thrust of 1.96 N. The initial pitch angle was determined as equal to 0.03 rad. In the simulation, the take-off cart tracks the aircraft’s position along the longitudinal axis of the launcher, without implemented control. With the assumed initial parameters, the aircraft lands on the cart in the fifth second of the motion, after which—according to the landing procedure—the system motion decelerates (Figure 22).

The implementation of VR software is invaluable for analyzing the process of UAV landing on a launcher cart. By visualizing the position of launcher elements and basic flight parameters, simulation parameters can be adapted in real-time to ensure a successful landing. Figure 23 displays selected images depicting key moments in the landing process, which are important for understanding the behavior of launcher system elements.

Table 4. The basic flight parameters corresponding to the data shown in Figure 23.

Time [s]	Distance [m]	Height [m]	Velocity [m/s]	Theta [rad]
0.5	10.5580	19.3775	21.2726	0.0476
3.0	67.0059	9.2986	24.9074	0.1651
4.0	92.1274	4.5337	26.1419	0.1248
5.0	118.3788	1.0202	26.7198	0.0597
5.2	123.7032	0.5271	26.7481	0.0456
5.4	129.0357	0.1094	25.8848	0.0317
5.5	131.0130	0.0167	16.1660	0.0374
6.0	138.0668	0.0071	13.8590	0.0515

summarizes the basic flight parameters corresponding to the data shown in Figure 23.

4. Discussion

The aim of the conducted numerical simulations was to analyze the takeoff and landing process of UAV in a fixed-wing configuration, which could not be achieved within the GABRIEL project. The authors wanted to analyze the takeoff process and determine the influence of initial conditions on its course, taking into account the limitations resulting from the design of the launcher, which were described in Section 2.1.

The UAV launcher should be designed to provide sufficient thrust to lift the UAV off the ground and achieve a flight speed that ensures continued flight. The parameters determining the UAV launch from the launcher include the thrust force, which must accelerate the aircraft to the required takeoff speed; the aircraft’s mass, which affects the required thrust and takeoff velocity, as well as the length of the takeoff runway; the pitch angle, which must generate the appropriate lift force required for takeoff; and the length of the launcher, which must be appropriate for the aircraft’s requirements, depending on its mass, wingspan, and other aerodynamic parameters.

The numerical analyses of the takeoff and landing of the UAV presented in Section 3 and their visualization in VR enabled verification of the developed mathematical model of the magnetic launcher system. The combination of the graphs of UAV and trolley motion parameters generated in MATLAB with a VR animation showing their mutual position and orientation allows for the analysis of the system’s motion. Additionally, it is possible to observe changes in the aircraft’s velocity vector and angle of attack, which provides a better understanding of the system’s dynamics.

The process of takeoff and landing of the UAV is dependent on initial conditions that determine the parameters of the system. In the analyzed project, the weight of the UAV taking off/landing must correspond to the parameters of the tracks so that the levitation force generated between the tracks and superconductors placed in the boxes of the starting cart is able to lift it. Therefore, the UAV used in the simulation had a weight of 1.5 kg. The key parameter is undoubtedly the length of the runway. However, in the analyzed magnetic launcher system, the launcher base (with rails mounted on it) is a modular structure, so the length of the launcher can be increased as needed. Therefore, in the conducted numerical simulations, the focus was on analyzing changes in other initial conditions affecting the dynamics of takeoff, namely the thrust force and pitch angle. As shown in Table 1, Table 2 and Table 3, where the moment of UAV launch from the launcher is highlighted in bold, depending on the initial pitch angle and thrust force of the starting UAV, the take-off velocity ranges from 24 to 57.6 m/s, and the aircraft detachment occurred between 0.1 and 2.7 s, while the length of the runway varies from 12.6 to 85 m. This demonstrates how small changes in initial conditions resulted in significant reductions in the takeoff run length.

The behavior of the UAV during takeoff is clearly visible in the VR application (Figure 19, Figure 20 and Figure 21). In the first simulation, the uncontrolled UAV without propulsion has to reach a significant takeoff speed, so after detaching from the launcher, it quickly transitions into a steep climb. In the second and third simulations, the angle of inclination of the UAV during takeoff is much smaller (around 0.03–0.04 radians) and the aircraft gains altitude much more slowly. In the third simulation, it almost slides off the launch pad, which could result in a collision with the launcher.

The conclusions drawn from the analysis of the UAV flight path in virtual reality confirm the MATLAB simulations visualized in Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. The UAV launch simulations from the launcher indicate that, in the absence of control, an important factor reducing the takeoff run and affecting the aircraft load is the adoption of a non-zero thrust force of the UAV during takeoff. The initial pitch angle of the take-off platform, (Figure 4a), of the launcher levitation sledge also has a significant impact on the launch process. As this parameter increases, the length of the take-off run decreases and the take-off speed decreases as well. This is demonstrated by the last two simulations, in which starting with the propulsion system turned on (at a low value of 3 N) caused faster detachment of the UAV from the launcher at a speed half as small (Figure 15). The trolley platform pitch angle value during UAV start-up is also important for operational reasons (Figure 16). An excessive pitch angle of the trolley may result in the trolley supports hitting the tracks and damaging them.

An interesting case was raised as part of simulation 3, where the angle of attack at take-off is higher than the angle of attack for UAV equilibrium conditions (Figure 18). Reducing the angle of attack during the first moments of independent flight leads to reducing the pitch angle (Figure 17) and lift force, hence, loss of UAV altitude (Figure 14), which may result in a collision with the take-off cart of tracks.

The conducted landing simulations turned out to be cumbersome in terms of selecting the initial conditions enabling the safe landing of an uncontrolled UAV on the take-off cart. They also revealed technical issues associated with safe aircraft landing. Above all, the cart-UAV connection structure should ensure sufficient stiffness and attenuation, so as to reduce relative aircraft linear and angular velocity relative to the cart. An aircraft touching down beyond the cart’s center of mass generates vibrations that have to be compensated for by launcher suspension attenuation.

The numerical analyses were partially validated in experimental studies described in works [18,24,25,26]. Based on the research described in [24], the calibration of the magnetic field parameters above the tracks of the magnetic launcher was carried out. In the experiment described in [25], the value of the magnetic force acting on a single support of the starting trolley was measured. The resulting levitation force as a function of the support height above the tracks was implemented into the simulation program. The damping coefficient of the levitation force was estimated based on data from a high-speed camera recording vibrations of the starting trolley support [18]. The recorded vibrations of the starting trolley support were reproduced in numerical simulations of the trolley dynamics model [26], which confirmed the correctness of the derived numerical model.

5. Conclusions

The launcher described herein is an innovative alternative to commercially available unmanned aerial vehicle launchers. The modularity is undoubtedly an advantage of the magnetic launcher design—the system consists of magnetic tracks and a platform from which the UAV takes off/lands. The spacing of boxes on the launch cart must be adapted to the spacing of the magnetic rails, while the length of the tracks can be increased by adding more modules. This allows for the mobility of the launcher and its application for various UAVs.

Deriving a system model required a thorough knowledge of mathematics, mechanics, electromagnetism and superconductors. The developed mathematical and numerical model allowed simulating cart motion at UAV take-off and landing. The results of conducted numerical simulations were implemented in VR software. They indicate that the design of the launcher discussed enables safe take-off, and that it is possible to safely land an aircraft on the launcher’s take-off cart with appropriately selected UAV and cart control algorithms.

The simulation results are essential for practical launcher operation. Based on them, it is possible to choose the best values of the UAV propulsion and the initial pitch angle during start-up to minimize the length of the take-off run and the load on the aircraft, while ensuring the safety and durability of the launcher. It should be emphasized that micro unmanned aerial vehicles are typically launched directly from the hand or using a very simple launcher similar to a rubber slingshot. The article presents the successive stages of research on an innovative aircraft takeoff and landing system using magnetic levitation. The developed research station, due to cost reduction, was built on a micro scale, hence the calculation example in the article was based on a micro-class UAV (weighing up to 2 kg). This test stand was a part of the GABRIEL project [15], which aimed to reduce exhaust emissions and noise of passenger aircraft by utilizing an electromechanical runway incorporating high-speed levitation rail technology using Inductrack technology. The project also aimed to land the aircraft on levitation sleds using the Inductrack runway [16]. Due to limitations of the Inductrack technology in small-scale devices and to reduce the cost of building the demonstrator, a test device using high-temperature superconductors to generate magnetic levitation forces was developed. This device included a levitation sled, a platform for the take-off and landing of small aircraft weighing up to 2 kg, a linear motor forcer, sled position and angular velocity sensors, a sled position and speed sensor along the magnetic track, motors for adjusting the position of the landing platform to the aircraft’s position, a computer, and an aircraft guidance system camera. A linear motor was used to power the levitation sleds, similar to the macro-scale device [17]. Of course, launching micro unmanned aerial vehicles does not require such complicated devices. However, the developed mathematical model can be easily scaled up to a much larger scale. In the future, the levitating runway can be used for launching electric motor gliders or general aviation aircraft. Classic takeoff of these aircraft from a runway at an airport consumes a lot of energy, significantly reducing the flight time and range. The advantage of the magnetic levitation launcher demonstrated in the study is the possibility of achieving relatively low speeds during takeoff.

As part of further research, we would like to develop the VR app so that it enables visualizing a larger number of system parameters, such as aircraft-cart reaction forces at take-off and landing or vectors and moments of aerodynamic forces. Controlling cart and aircraft motion under variable environmental conditions, so as to avoid a collision is also a significant problem. These issues constitute the origin and interpretation for future studies.