2.1 Kinematics equation of UAV launch process

From a kinematic viewpoint, the launch operation can be described as accelerating the UAV and trolley along a launch frame under the action of the engine thrust and traction force of a traction rope while overcoming friction, aerodynamic forces including lift and drag, and the force of gravity. The traction force is provided by the action of compressed gas on the cylinder piston rod and transmitted by an accelerating pulley block. A schematic diagram of the forces during the launch process is shown in Fig. 1.

Figure 1. Force analysis diagram of the UAV acceleration process.

Considering the UAV and trolley as point masses, one can derive the equation of motion of the UAV acceleration process as

(1) \begin{align} {m_Z}\frac{{{d^2}x}}{{d{t^2}}} = {F_t} + {T_e}\cos (\alpha ) - ({m_Z}g - L\cos (\gamma ))\sin \gamma - {F_f} - D \end{align}

where ${F_t}$ is the traction in the rope, ${T_e}$ is the UAV engine thrust, $L$ is the aerodynamic lift, $D$ is the aerodynamic drag, ${F_f}$ is the friction between the trolley and the sliding rail, ${m_Z}$ is the total mass of the trolley and the UAV, $\;x$ is the displacement of the trolley and the UAV, $\;\alpha $ is the installation angle of the UAV and $\gamma $ is the launch angle of the device.

The aerodynamic force and friction between the trolley and sliding rail of the launcher can be calculated as:

(2) \begin{align} L = \frac{1}{2}S{C_L}\rho {v^2} \end{align}

(3) \begin{align}D = \frac{1}{2}S{C_D}\rho {v^2}\end{align}

(4) \begin{align}{F_f} = \mu ({m_z}g\cos \gamma - L)\end{align}

where $S$ is the wing area, ${C_L}$ is the lift coefficient, ${C_D}$ is the drag coefficient, $\rho $ is the density of air, $v$ is the speed of the UAV and $\mu $ is the friction coefficient of the system.

2.2 Dynamic model of the accelerating pulley block

As shown in Fig. 2, the load force on the cylinder piston rod is transmitted to supply the driving force on the movable pulley assembly, UAV and pulley during the launch process. The accelerating pulley block, which includes a fixed pulley and a movable pulley^{(Reference Wei, Xiaoping, Ming and Yang13)}, increases the pneumatic launch speed and stroke of the UAV because the stroke and end speed of the cylinder as an actuator are limited. The free end of the steel wire rope is connected to the pulley, while the other end wraps around the reversing fixed pulley, which is wound with the movable and fixed pulleys in turn, while the end is fixed.

Figure 2. Schematic diagram of the accelerating pulley block.

When the cylinder starts to act and provides a driving force for the launch, the accelerating pulley block acts to increase the speed and distance. Simultaneously, the transmission of the driving force is completed. To analyse the relationship between the traction force of the rope acting on the UAV and pulley and the driving force of the cylinder, the movable pulley assembly is considered as the moving body, and several reasonable assumptions should be made before the dynamic analysis:

1. (1) The traction rope does not deform during the launch process, and its own weight and rigid resistance can be ignored;

2. (2) The rotation speed of the pulley at a given time is identical to the linear speed of the wire rope contact point;

3. (3) The moving pulley assembly and cylinder piston rod move synchronously; i.e., their speeds are identical.

The dynamic equation for the moving pulley group can thus be expressed as

(5) \begin{align}{F_z} = {F_t} + \sum\limits_{i = 1}^{{n_h} - 1} {{F_i} + {f_h} + {m_h}\frac{{{d^2}s}}{{d{t^2}}}} \end{align}

where ${F_z}$ is the load force of the cylinder piston rod, ${F_t}$ is the rope traction, ${x_1}$ is the displacement of the movable pulley assembly and the cylinder piston rod, ${F_i}$ is the force of the $i - th$ wire rope in the pulley block, ${m_h}$ is the mass of the movable pulley assembly, ${f_h}$ is the friction of the movable pulley assembly force and ${n_h}$ is the pulley block multiplication factor.

The forces on the winding-in side and winding-out side of the same pulley are not identical, being related as

(6) \begin{align}{F_i} = {F_{i = 1}} + {f_{hi}}\end{align}

where ${f_{hi}}(i = 1,2, \cdots n - 1)$ is the friction force of the rope on the $i - th$ pulley. When $i = 1$ , ${F_0} = {F_t}$ , one has

(7) \begin{align}{F_1} = {F_t} + {f_{hi}}\end{align}

According to the preceding analysis, the driving force for the rotation of the pulley is generated by the friction between the wheel and the rope. The rotation of the pulley generates a torque. Combined with the friction at the pulley bearing, the following can be obtained:

(8) \begin{align}{F_z} = {K_1}{F_t} + {K_2}\frac{{{d^2}x}}{{d{t^2}}} + {\mu _h}{m_h}g\end{align}

(9) \begin{align}{K_1} = n + \frac{{{\mu _1}{r_c}}}{{{r_h} - {\mu _1}{r_c}}}\sum\limits_{i = 0}^{n - 2} {(n - 1 - i){{(1 + \frac{{{\mu _1}{r_c}}}{{{r_h} - {\mu _1}{r_c}}})}^i}} \end{align}

(10) \begin{align}{K_2} = {m_h} + \frac{{n{I_p}}}{{{r_h}({r_h} - \mu {r_c})}}\sum\limits_{i = 0}^{n - 2} {(n - 1 - i){{(1 + \frac{{{\mu _1}{r_c}}}{{{r_h} - {\mu _1}{r_c}}})}^i}} \end{align}

(11) \begin{align}{I_p} = {m_p}(r_n^2 - r_c^2)\end{align}

where ${\mu _h}$ is the friction factor of the moving pulley block, ${\mu _1}$ is the friction factor corresponding to the rotation of the pulley bearing, ${I_p}$ is the inertia of the pulley, ${r_h}$ is the radius of the pulley, ${r_c}$ is the radius of the bearing and ${m_p}$ is the mass of a single pulley.

2.3 Dynamic analysis of the pneumatic transmission system

In a pneumatic launch, the pneumatic system mainly provides energy to speed up the UAV. The entire working process is a short-term thermodynamic process that is both complex and diverse. To facilitate this theoretical analysis, the gas is considered to be ideal. The process is considered to be quasi-static and adiabatic. Therefore, the work done by the pneumatic system during the entire launch process can be divided into the dynamic process of constant-volume cylinder deflation and high-speed cylinder movement, thus the mathematical model for the launch process can be obtained.

1. (1) Mathematical model of the gas cylinder deflation process

The gas cylinder deflation diagram is shown in Fig. 3.

Figure 3. Schematic diagram of the gas cylinder deflation model.

We assume that the volume of the gas cylinder is $V$ and that the container has been filled with compressed air before deflation, and use the adiabatic venting energy equation for a limited volume^{(Reference Li and Pei14–Reference Jian Fan16)},

(12) \begin{align} - \kappa RTdM = \kappa pdV = Vdp \end{align}

For the gas cylinder, $dV = 0$ and $dM = {Q_m}dt$ . The equation for the change in pressure during the gas cylinder deflation process with time can be expressed as

(13) \begin{align}dt = \frac{{dM}}{{{Q_m}}} = - \frac{{Vdp}}{{\kappa RT{Q_m}}}\end{align}

(14) \begin{align}\frac{{dp}}{{dt}} = - \frac{{\kappa RT{Q_m}}}{V}\end{align}

where $\kappa $ is the ratio of specific heats, $R$ is the gas constant, $T$ is temperature in Kelvin, $M$ is the gaseous mass, $\rho $ is the gas density, $V$ is the cylinder volume, ${Q_m}$ is the gas mass flow rate and $t$ is time.

1. (2) Mathematical model of the cylinder work process

Figure 4 shows a schematic diagram of the work process of the cylinder. ${X_{10}}$ is the length of the cylinder with rod cavity clearance, ${X_{20}}$ is the cylinder without rod cavity clearance length, ${S_s}$ is the cylinder actuation stroke, ${X_s}$ is the cylinder stroke at moment $t$ and ${S_a}$ is the cylinder exhaust area.

Figure 4. Schematic diagram of the cylinder operating process.

Suppose that ${P_s}$ is the air source pressure, ${T_s}$ is the temperature, ${A_1}$ , ${P_{10}}$ , ${T_{10}}$ , ${V_{10}}$ and ${A_2}$ , ${P_{20}}$ , ${T_{20}}$ , ${V_{20}}$ are the area, initial pressure, temperature and volume of both sides of the cylinder; ${T_1}$ , ${P_1}$ , ${V_1}$ and ${T_2}$ , ${P_2}$ , ${V_2}$ are the parameters at moment t.

According to the relevant theory of a pneumatic transmission system, the equation for the change in the pressure in the cylinder rod cavity can be obtained as

(15) \begin{align}\frac{{d{p_1}}}{{dt}} = \frac{{\kappa R{T_s}{Q_{m1}}}}{{{V_1}}} - \frac{{\kappa {p_1}}}{{{V_1}}}\frac{{d{V_1}}}{{dt}}\end{align}

The equation for the pressure change in the rodless cavity of the cylinder can be expressed as

(16) \begin{align}\frac{{d{p_2}}}{{dt}} = \frac{{\kappa {p_2}}}{{({S_s} + {X_{20}} - {X_s})}}\frac{{dX}}{{dt}} - \frac{{\kappa R{T_2}{Q_{m2}}}}{{{A_2}({S_s} + {X_{20}} - {X_s})}}\end{align}

3.1 Experimental methods

Based on a virtual prototype of the pneumatic launch system, test pieces of the parts were processed and assembled. A generator powered the entire device, and air compressors stored energy for gas cylinders mounted outside the launcher. A cylinder was installed into the device to perform the pull-back action. A five-fold accelerating pulley block device was used to increase the speed and distance. The pneumatic system was built, and a Programmable Logic Controller (PLC) was applied for integrated control of signal collection. During the pneumatic launch, the pressure signal in the pneumatic transmission system was collected by the PLC, and the air compressor was controlled to store the energy in the storage cylinder. When the set working pressure was reached, the pneumatic servo valve was controlled to open to start the cylinder operation and perform the ground test of the pneumatic launch process. The pneumatic launch test is shown in Figs. 5 and 6.

Figure 5. Pneumatic catapult structure diagram.

Figure 6. Pneumatic launch test chart.

The ground test of the pneumatic launch system was performed without aerodynamic loads. Equivalent mass blocks were used to simulate UAVs of different masses. By measuring the launch device angle, the height of the mass as it leaves the device and location of the mass block, we calculated the maximum launch speed that the launch device could achieve. The mass of the UAV and pneumatic parameters were adjusted by changing the mass and air compressor parameters to satisfy the index requirements. Several tests were conducted, and the influence of different parameters on the dynamic characteristics of the pneumatic launch device was studied.

The core formula for the experimental system measurements can be obtained as follows:

(17) \begin{align}\left\{ {\begin{array}{*{20}{l}}{H = {v_0} \cdot {t_1} \cdot \sin \theta - \frac{1}{2}g{t_1}^2}\\{X = {v_0} \cdot {t_1}\cos \theta }\end{array}} \right.\end{align}

The launch time can be expressed as

(18) \begin{align}{t_1} = \sqrt {\frac{{2(X \cdot \tan \theta - H)}}{g}} \end{align}

Thus, the maximum launch velocity can be expressed as

(19) \begin{align}{v_0} = \frac{{X\sqrt g }}{{\cos \theta \cdot \sqrt {2(X \cdot \tan \theta - H)} }}\end{align}

where H is the height of the mass from the frame, g is the acceleration due to gravity, ${\theta}$ is the launch angle, X is the distance between the landing point of the mass and the frame, t ₁ is the landing time of the mass and v ₀ is the launch velocity.

3.2 Model verification

Three sets of experiments were performed using the pneumatic launcher, and the test results were compared with the simulation results. The parameters studied were as follows: UAV installation angle α = 2°, launch angle ${\gamma}$ = 8°, cylinder outer diameter D_c = 250mm, cylinder piston rod diameter d_c = 50mm, cylinder volume V = 80L, inflation pressure P = 0.39MPa, pulley mass m_c = 8kg, pulley block moment of inertia J_c = 0.02 kg ${\cdot}$ m ² and friction coefficient μ = 0.01.

The comparison results are presented in Table 1.

Table 1. Comparison of simulation and test results

As shown in Table 1, the theoretical results agree well with the test results. The errors are all within 5%, providing that the established dynamic model is relatively accurate and can be used for simulation analysis of UAV pneumatic launch dynamics.

4.1 Effect of cylinder pressure

The working pressure of the gas cylinder of the pneumatic transmission system affects the dynamic characteristics of the launch system. To analyse the influence of the pneumatic transmission system-related parameters on the launch performance, the working pressure of the gas cylinder was set to 0.19MPa, 0.39MPa and 0.59MPa. Other system parameters were taken as fixed values, specifically: the mass of UAV was 20kg, the launch angle was 8°, the pulley block moment of inertia was 0.02kg ${\cdot}$ m ², the cylinder volume was 80L and the cylinder action area was 0.042m ². We performed the simulation using the co-simulation model. The speed and acceleration results are shown in Figs. 7 and 8.

Figure 7. The influence of the air tank pressure on the take-off speed.

Figure 8. The influence of the air tank pressure on the take-off overload.

As shown in Figs. 7 and 8, under identical launch distance conditions, the cylinder pressure greatly affected the launch process, including the take-off speed and overload. The maximum take-off speed of the UAV increases with increasing inflation pressure, and the upward trend of the speed increases. The reason is that, during the launch process, the pneumatic actuator must overcome all external loads. With increasing cylinder pressure, the pneumatic actuator provides a relatively strong force, which increases the take-off speed and overload of the UAV. The key values describing the effect of the changes in cylinder pressure on the launch performance are presented in Table 2.

Table 2. Influence of cylinder pressure

Table 3. Influence of other parameters

4.2 Influence of other parameters

We repeated the steps described in Section 3.1 to simulate and analyse the UAV mass, launch angle, cylinder volume, pulley block moment of inertia and cylinder action area. The parameters were changed to study the effect on the dynamic response of the pneumatic launch system, as presented in Table 3.

The results presented in Table 3 show that, during the UAV launching process, the take-off speed and take-off overload of the UAV increased with increasing inflation pressure, volume of the gas cylinder and effective area of the cylinder; i.e., they show a positive correlation, but decreased with increasing mass of the UAV and moment of inertia of the pulley block, revealing a negative correlation.

4.3 Sensitivity analysis of system parameters

Based on the preceding analyses of the effects of key parameters on the launch performance, we conducted a sensitivity analysis of the key parameters, which can efficiently evaluate the effects of their variation on the launch speed and overload. To further explore the impact of the analysed system parameters on the performance of the launch system and lay the foundation for subsequent research on the matching characteristics of the UAV and launcher, the optimal Latin hypercube design method in the Design of Experiments (DOE) component of Isight⁽¹⁹⁾ was used for the analysis^{(Reference Munk, Auld, Steven and Vio20)}.

For the sensitivity analysis, the UAV mass, launch angle, moment of inertia of the accelerating pulley block, volume of the gas cylinder, inflation pressure of the gas cylinder and effective area of the cylinder were used as the design variables, and the launch speed and overload as the responses. Thus, the percentage sensitivity is expressed as

(20) \begin{align}{S_{ri}} = \frac{{{S_i}}}{{\sum\limits_{i = 1}^{{n_1}} {\left| {{S_i}} \right|} }} \times 100\% \end{align}

(21) \begin{align}{S_i} = \frac{{f({x_1},{x_2}, \cdots ,{x_i} + \Delta {x_i}, \cdots ,{x_n})}}{{\Delta {x_i}}} - \frac{{f({x_1},{x_2}, \cdots ,{x_i} + \Delta {x_i}, \cdots ,{x_n})}}{{\Delta {x_i}}}\end{align}

where ${S_{ri}}$ is the percentage sensitivity, ${S_i}$ is the sensitivity of each design variable change and ${n_1}$ is the total number of design variables.

The results provide a Pareto diagram of the influence of these design variables on the launch speed and overload. A positive, blue value indicates a positive effect, while a negative, red value represents a negative effect. A positive effect shows that the index increases when the design variable is increased, and a negative effect that the index decreases when the design variable is increased. The magnitude of the value indicates the degree of the influence on the system.

As shown in Figs. 9 and 10, the UAV characteristics, the moment of inertia of the pulley block and the inflation pressure of the gas cylinder are the most important parameters that influence the take-off speed and launch overload of the launch system. Increasing the mass or moment of inertia of the UAV has a negative effect on the take-off speed and launch overload, while increasing the pressure has a positive effect.

Figure 9. Pareto chart of speed.

Figure 10. Pareto chart of acceleration.

5.1 Analysis of matching characteristics of launch system parameters

According to the parameter sensitivity analysis described in Section 3, it can be concluded that the three key parameters that affect the launch performance are the parameters of the UAV system, the mechanical structure parameters and the parameters of the pneumatic transmission system of the launch system. To study the maximum matching area of the three key parameters under the constraints of the launch take-off safety criterion, according to the relevant safety criteria of UAV launches in literature^{(Reference Lucas21–Reference Li22)}, the results can be summarised as follows:

(22) \begin{align} \left\{ {\begin{array}{*{20}{c}}{{V_{\max }} \le \sqrt {\frac{{2T\cos (\alpha + \gamma )}}{{\rho {S_{ref}}g(\alpha )\cos (\angle f(\alpha ) + \gamma )}}} }\\{{V_{\min }} \ge \frac{{ - 0.5m\sin \gamma + \sqrt \Delta }}{{\rho {S_{ref}}g(\alpha )\cos (\angle f(\alpha ) + \gamma )}}}\end{array}} \right.a \end{align}

(23) \begin{align}{V_{\min }} \lt V \lt {V_{\max }}\end{align}

where

(24) \begin{align}\Delta = 0.25{m^2}{\sin ^2}\gamma - 2\rho {S_{ref}}g(\alpha )\cos (\angle f(\alpha ) + \gamma )(T\sin (\alpha + \gamma ) - mg)\end{align}

(25) \begin{align}\cos \angle f(\alpha ) = \frac{{0.079\alpha + 0.393}}{{\sqrt {{{(0.0036\alpha + 0.024)}^2} + {{(0.079\alpha + 0.393)}^2}} }}\end{align}

(26) \begin{align}a \le 0.5{V_{\min }}\end{align}

In these formula, ${V_{\max }}$ and ${V_{\min }}$ are the maximum and minimum speeds of the UAV launch and take-off, respectively, $T$ is the engine thrust, $\alpha $ is the installation angle of the UAV on the launch system, $\gamma $ is the launch angle, $\rho $ is the standard atmospheric pressure density, ${S_{ref}}$ is the aircraft wing area and $a$ is the launch overload of the UAV. Hence, the speed range of the UAV launch should satisfy $21m/s \lt V \lt 31m/s$ , and the launch overload should satisfy $a \le 10.5g$ .

The maximum working pressure of a pneumatic actuator is generally 0.8MPa, and the moment of inertia of pulley blocks should not be excessive due to limitations on their size and weight. Combined with engineering experience, the technical index of the launch mass of a UAV is $M = (5\sim 80)kg$ , the working pressure of the pneumatic actuator is $P = (0.1\sim 0.8)MPa$ and the moment of inertia of the pulley blocks is kg ${\cdot}$ m ². A step size is adopted to perform the batch simulation. Based on the boundary conditions of the iterative simulation calculation from the safety criterion, a three-dimensional characterisation diagram of the parameter matching value set for the UAV system and pneumatic launch system that satisfies the safety criterion at launch can be obtained; the results are shown in Figs 11–14.

Figure 11. Main UAV launch system parameters.

The batch simulation results show that the UAV can be safely launched when the matching values of the system parameters $(M,P,RI)$ are located inside the envelope shown in the figures. A closer parameter matching value corresponds to the light-colour area with higher take-off speed and overload of the UAV.

Figure 12. Main UAV launch system parameters.

Figure 13. Main UAV launch system parameters.

In addition, Figs 11 and 12 show, that under the safety criteria of the UAV launch take-off speed and overload, the mass $M$ of the UAV increases with a decrease in the rotational inertia ${I_p}$ of the pulley block and an increase in the inflation pressure $P$ of the gas cylinder. Thus, reducing the moment of inertia of the pulley block and increasing the charging pressure of the gas cylinder can improve the maximum take-off mass of the UAV, while also expanding the application range of the pneumatic launch system. In the vertical view shown in Fig. 13, the abscissa represents the range of mass $M$ of the UAV while the ordinate represents the range of the inflation pressure $P$ . When changing the inflation pressure, the range of the UAV mass change increases, indicating that the inflation pressure greatly affects the matching range of the UAV mass. In the right view shown in Fig. 14, changing the moment of inertia of the pulley block ${I_p}$ hardly affects the UAV mass matching area. Therefore, when top-level parameter matching design is performed for a UAV pneumatic launch system, to enable the launch system to be applied to more massive UAVs, the focus should be on the design and analysis of the inflation pressure.

Figure 14. Main UAV launch system parameters.