Autonomous Tilt Rotor Stabilized Plimp Hybrid Airship Unmanned Aerial Vehicle

Abstract

Stabilizing the remotely operated hybrid VTOL tilt rotor unmanned aerial vehicle that has an envelope of gas particularly installed to provide hydrostatic buoyancy force to generate lift, with a payload bay equipped with sensors and electronic components attached to the envelope and having a pair of wings extended on both the sides of the bay to generate lift through motion as well act as a stabilizing surface, and the thrusters pivoted at wing tip would be tilted with powerful servo motors to provide vertical takeoff and landing capabilities. This manuscript deals with stabilizing the undamped forces and moments using automatic control system. Mathematical modeling of equations would be framed based on the obtained kinematics of flight that would be converted to autopilot codes in MATLAB. The MATLAB results further determine the stability of the designed tilt rotor Plimp unmanned aerial vehicle.

1 Literature Review

Moutinho et al. [1] use Lyapunov’s theory to analyze nonlinear system stability. Tests have been done to verify the nonlinear performance of controller and thereby correcting the disturbances and errors.

Andan et al. [2] show that the lift force would be three times increased for the airship with wing structure at a positive angle of attack. 20 to 40% of increment in drag occurs with winged airship. Cook et al. [3] explain the various lateral-directional flight modes of the Plimp including modes like sideslip subsidence, yaw subsidence, and oscillatory roll pendulum, also comparison was made between the estimated models and existing airship for various speeds.

DeLaurier et al. [4] performed analysis to develop stability of airships for the non-neutral net buoyancy conditions and non-coincident mass and volumetric centers conditions. Li et al. [5] proposed a method for simulation of airships in nonlinear dynamics. Both the model of the statics and dynamics of air were framed. Wang et al. [6] used CFD and Fourier analysis to obtain the stability derivatives.

Ceruti et al. [7] describe the optimization of airship that consist of two semi-ellipsoids, and axis ratios were altered for the same. The various parameters to optimize were volume, dimension of the tail, ratio between the vertical and the lateral semi-axis, the percentage coverage of photovoltaic films on surface of the top, and the ratio between the longitudinal and the lateral semi-axis. Andan et al. [8] presented the results of a numerical study of aerodynamic parameters for a wingless as well as a winged airship. For various angles, the net force coefficients and moment coefficients have been calculated.

2 Introduction

The Plimps are the flying devices that can be described as an aircraft with plummet-proof which has lifting capability of an airplane, the thrust control mechanism of a helicopter, and the lift due to buoyancy forces of a Plimp [9]. Egan Airships has designed their eight-passenger Model J. Plimp which are new type of airship, with a combination of helium envelope and dynamic rotors enabled wings, allowing it to perform operations like hovering, dipping, ascending, banking, or spinning [10]. Even at times of engines off mode, it would simply float and glide smoothly to the land. Having VTOL capability, it could lift off from anywhere without runway and land anywhere without runway [11]. The Plimp was initially designed by Daniel P. Raymer, who is a famous expert in the discipline of aircraft conceptual design and aircraft design engineering [12]. Advantage of Plimp being that the aircrafts are noisy, Plimps are not.

3 Design

The pimp would be having a hull, structural frame, thrusters with servos to pivot, the H-tail, and the flight control with an embedded system. The pressurized inflatable envelope would be filled with the suitable gas that is lighter than air that should provide maximum static lift in air for the unmanned aerial airship. Though hull portion of the airship could be shaped with various geometries, the ellipsoidal shaped hull would be preferred for this tilt rotor stabilized hybrid Plimp unmanned aerial vehicle due to high efficiency with less surface area to volume ratio and thereby power consumption could be minimized. This would improve the endurance of the UAV. The tilt rotors would vector the thrust force in necessary directions. Vectoring the thrust components would ease the Plimp to correct its flight path and provide necessary stability in any particular axes. The weight due to gravity, the lift due to aerodynamics, the aerostatic lift, and the thrust vectored lift would be considered as the major force that are acting on this unmanned airship. Among these forces, the aerostatic lift would be given by the differences between the force due to buoyancy and the weight of gas displaced (Fig. 1). Consider volume (V) of the envelope and the density $\left( \rho \right)$, the aerostatic lift would be given by

$${\text{Lifting}}\,{\text{force}}\left( {L_{f} } \right) = V\left( {\rho_{{{\text{air}}}} - \rho_{{{\text{gas}}}} } \right)$$

(1)

Fig. 1

Free body diagram of takeoff [13]

$$F = m\left( {\frac{{{\text{d}}v}}{{{\text{d}}t}}} \right) = {\text{Thrust}}\left( {F_{{\text{t}}} } \right) + {\text{Buoyancy}}\left( {F_{{\text{b}}} } \right) - {\text{Weight}}\left( W \right) {\text{Drag}}(F_{{\text{d}}} )$$

(2)

where

$${\text{Drag}}\left( {F_{{\text{d}}} } \right) = 0.5\,C_{{\text{D}}} \rho_{{{\text{air}}}} Av^{2}$$

(3)

$C_{{\text{D}}}$: Drag coefficient and depends on shape,

$\rho_{{{\text{air}}}}$: Air density,

A: Area (cross-sectional),

v: Velocity.

The design of envelope has the major impact on the stability, so the necessary design considerations were adopted while designing them. GNVR shape has been chosen to construct the airship. The geometric profile of the airship is given in Fig. 2.

Fig. 2

GNVR profile for the airship [14]

Choice of gas being used I the envelope is also most essential part of the design. So for the comparison, let’s tabulate the various gases commonly used along with their densities and molar mass (Table 1).

Table 1 Density and molar mass of gases at standard conditions

3.1 Material Selection

Factors affecting the material selection for the envelope; would be the price, the sturdiness, the stress sustaining capability during various flight conditions, including the infiltration of the lifting fluid (gas) [15]. High strength to weight ratio, high tear resistance, resistance to the environmental degradation, and low permeability to LTA gases are the basic material property for choosing such inflatable structure [16, 17]. The biaxial-oriented polyethylene terephthalate in short known as BOPET is selected for inflatable structure material. The metalized BOPET, also known as Mylar, is cheaper than the normal polyurethane. But the studies suggested that it is susceptible to gas (helium) leakage [18]. If the Mylar gets punctured, it would wear out rapidly than the polyurethane material [19].

3.2 Stability of Airship

Considering the stability of airships, it could be defined through the classification by static and dynamic. The classification of stability considered during the no powered flight condition would be termed as static stability. The phenomenon of return back to its original position despite of disturbances defines this condition [20] (Table 2).

Table 2 Axes of airship and conventional symbols related to them

In general, airships are statically unstable in yaw. Effect of dynamic stability comes in the picture when the airstream flow passes through the control surfaces. Though the stability of airplane and airship seems to have similar classification of stabilities, one of the major differences being that the stabilities in the case of airplanes is associated with one another, but in the case of airships, they being independent of each other. In steady flight, pitch stability, yaw stability, and roll stability are the various stabilities involved.

Assumptions that need to be made while performing the derivation of the stability parameters:

• (1) The net weight of the body remains constant.

• (2) Considering the accessional force to remain constant.

• (3) Fixed center of gravity as well as center of buoyancy.

• (4) The controls remain in neutral.

• (5) Constant velocity.

• (6) No changes in the form of airship [20].

3.3 The Various Forces and the Various Moments Acting on the Plimp

Consider Plimp that flying along the horizontal path, such that the flight path makes an angle of $0^{^\circ }$ with the longitudinal axis, then the various forces and the moments acting on the Plimp would be (see Fig. 3) [20].

Fig. 3

Axes and angles in positive direction [20]

• (1) Forces

$L_{0} = {\text{Lift of inflating gas acting through center of buoyancy}}, G$.

$W = {\text{Total weight of dead and live loading}},\,{\text{acting through center of gravity}}, M$.

$R = {\text{Resistance of envelope and appendages, acting through center of pressure}}, P$.

$T = {\text{Propeller thrust, acting parallel to axis of envelope at distance }}o {\text{below}} M$.

• (2) (2) Moments about M

$MomentL_{0} = L_{0} \times 0 = 0$.

$Moment W = W \times 0 = 0$.

$Moment thrust - resistance couple = T\left( {c + d} \right)$.

Condition for static equilibrium and keeping constant velocity

$$L_{0} = W$$

(4)

$$R = T$$

(5)

During the flying condition of Plimp on an even keel, the moments due to thrust force and resistance force would be unbalanced; this would nose up the Plimp. To handle such phenomenon when Plimps are full of gas, are regularly trimmed a few degrees nose heavy. In case of gust, disturbances in the longitudinal axis would give rise to a slight tilt from the horizontal plane, few cases could be observed and described as Table 3.

Table 3 Six cases depending on the static state of Plimp with the course of inclination [20]

From Fig. 4, the forces, the lever arms, and the moments, for the cases one to six are noted:

Fig. 4

Forces on airship in horizontal flight [20]

• (1) Forces

  • $L_{{\text{g}}} = {\text{Lifting force of fluid}}$.

  • $W = {\text{The Net Weight acting due to gravity}}$.

  • $F_{{\text{e}}} = {\text{Resultant air force acting on the envelope}}$.

  • $L_{{\text{s}}} = {\text{Lift of tail surface}}$.

  • $F_{{\text{s}}} = {\text{Resultant force acting on the tail surfaces}}$

  • $T = {\text{The thrust of Propeller}}$.

  • $L_{{\text{e}}} = {\text{Vertical component of the forces due to motion acting on the envelope}}$.

  • $t = {\text{The horizontal component of the thrust produced by propeller }}$.

  • $R_{{\text{e}}} = {\text{Horizontal component of the forces due to motion acting on the envelope}}$.

  • $R_{{\text{s}}} = {\text{Drag of the tail surface}}$.

  • $L_{{\text{t}}} = {\text{The vertical Component of thrust produced by propeller }}$ [20].

• (2) Leaver Arms

  • $W = K\sin \left( {\alpha \pm \theta } \right)$

  • $L_{{\text{g}}} = 0$.

  • $T = \left( {c + h} \right)$.

  • $F_{{\text{s}}} = a\left( {{\text{assuming }}F_{{\text{s}}} ,{\text{perpendicular tothe surfaces}}} \right)$.

  • $L_{{\text{s}}} = a \times \cos \left( {\alpha \pm \theta } \right)$.

  • $R_{{\text{s}}} = a \times \cos \left( {\alpha \pm \theta } \right)$.

  • $F_{{\text{e}}} {\text{ would vary with the position of }}P,{\text{ which in turn would depend on }}\theta$.

  • $L_{{\text{e}}} = b \times \cos \left( {\alpha \pm \theta } \right)$.

• (3) Moments

  • Moment due to weight is $W h\sin \left( {\alpha \pm \theta } \right).$

  • Moment due to propeller thrust is $T \left( {c + h} \right).$

  • $F_{{\text{e}}}$ tend to turn the complete Plimp in the positive direction about $M$ due to the increased pressure below the hull. This phenomenon is assisted by reducing the pressure in bottom surface of the tail. The forces acting below the nose of the UAV and the tail of the UAV would be in opposite direction. As the nose force is to some extent is greater than the tail force, there is a difference in force, which will be known as the dynamic lift of hull. Despite of the difference, both the forces have same direction of rotation, and the resultant moment caused is dynamic upsetting moment, denoted by $M_{{\text{e}}}$.

  • Moment due to the tail surface, denoted by $M_{{\text{s}}}$ opposes this dynamic upsetting moment. $M_{s} = L_{s } \alpha {\text{cos}}\left( {\alpha \pm \theta } \right) + R_{s } \alpha \sin \left( {\alpha \pm \theta } \right).$

3.4 Plimp Stability from Designer Point of View

• Plimps are very stable about their lateral axis.

• Design inputs needed for Plimps to provide longitudinal stability.

• In yaw, Plimps are statically unstable, only pilots can handle this through their rudders.

3.5 Requirements of the Hybrid VTOL Plimp Airship Unmanned Aerial Vehicle

• Payload bay that can carry payload along with the essential electronic components.

• The primary lifting device that would be integrated to the payload bay and installed to provide hydrostatic buoyancy.

• The secondary lifting device integrated to the fuselage and installed to provide dynamic lift through movement of the secondary lifting device through the air.

• The thrust system equipped to generate thrust, the thrust system integrated to the secondary lifting device and it rotates together about an axis that is aligned with the spar of the wing [21].

• The tail system that could be pivoted upon the tail boom to counteract the unwanted forces and moments produced by the tilt rotors [22].

Through the requirements defined above, the hybrid VTOL airship unmanned aerial vehicle with H-tail has been designed using the modeling software CATIA. The various three-dimensional geometric views of the unmanned aerial vehicle are shown in Figs. 5 and 6.

Fig. 5

Isometric, top, side, and front views of Plimp during forward tilt of thrusters

Fig. 6

Isometric, top, side, and front views of Plimp during upward tilt of thrusters

4 Calculations

4.1 Weight Estimation

See Table 4.

Table 4 Weight estimation of the UAV

4.2 Airfoil Data for the Wing

The airfoil used in this UAV is Bell A821201 (23%) FX-66-H-60, because most of the thrust vector is going to be away from the chord line. This airfoil has flat bottom surface and streamlined upper surface which helps UAV to float stably irrespective of thrust direction.

4.3 Propeller Data

Propeller: Radius 12.7 cm.

$${\text{Area}} = \pi r^{2} = 0.507\,{\text{m}}^{2}$$

(6)

4.4 Different Phases of Flight

Hovering

Calculations of various performance parameters during hovering are given by [23].

Generally, the thrust produced by the motors and the propellers combination should be sufficient to lift the total weight including payload of the UAV

$$T = 2\rho A(V + V_{i} )V_{i} = 3.3\,{\text{kg}} = 32.34\,{\text{N}}$$

(7)

$$V_{{\text{h}}} = \sqrt {\frac{W}{2\rho A}} = {16}.{14}\,{\text{m/s}}$$

(8)

$$\Omega = \frac{2 \pi N}{{60}} = {1278}.{62}\,{\text{rad/s}}$$

(9)

$$V_{{\text{h}}} = \Omega \,R\sqrt {\frac{{C_{T} }}{2}}$$

(10)

Thrust of a single BLDC motor with the propeller attached is given by

$$T = C_{{\text{T}}} \rho \left( {\Omega R} \right)^{2} \,\,A = {32}.{4265}\,{\text{N}}$$

(11)

Torque of a single BLDC motor with the propeller attached is given by

$$Q = C_{Q} \rho \left( {\Omega R} \right)^{2} \,\,AR = 0.5122\,{\text{Nm}}$$

(12)

$${\text{F}}.{\text{O}}.{\text{M}} = \frac{{C_{{\text{T}}}^{3/2} }}{{\sqrt 2 C_{{\text{Q}}} }}$$

(13)

$$P = Q\,\,\Omega = 654.89\,{\text{Nm/s}}$$

(14)

Climbing

Calculations of various performance parameters during climbing are given by [23]

$$D = 0.5\,\rho \left( {V + V_{i} } \right)^{2} A_{{\text{B}}} C_{{{\text{DB}}}} = {8}.{9484}\,{\text{N}}$$

(15)

$$T = D + W = 41.2936\,{\text{N}}$$

(16)

Therefore,

$$\rho A\left( {V + V_{i} } \right)2 V_{i} = 0.5 \rho \left( {V + V_{i} } \right)^{2} A_{{\text{B}}} C_{{{\text{DB}}}} + W$$

(17)

$$Q = 4A\left( {V + V_{i} } \right) \,\,\omega R^{2} = 1.02097\,{\text{Nm}}$$

(18)

$$Q\left( {\Omega - \omega } \right) = T\left( {V + V_{i} } \right)$$

(19)

$$P = Q\,\,\Omega = 1305.4327\,{\text{Nm}}$$

(20)

where

$V_{c}$ is climbing velocity,

$D _{ }$: Drag generated,

$AB_{ }$ is the area of the propeller,

$C_{{D_{B} }}$ is the drag coefficient due to the propeller.

Forward

Calculations of various performance parameters during forward are given by [23]

$$D = 0.5\,\,\rho V_{{\text{R}}}^{2} A_{{\text{B}}} C_{{{\text{DB}}}} = {34}.{78}0{2}\,{\text{N}}$$

(21)

$$\begin{aligned} \tan \delta & = \frac{D \cos \varepsilon }{{D \sin \varepsilon + W }}, \\ \delta = & 46.5301 \\ \end{aligned}$$

(22)

$$T^{2} = D^{2} + W^{2} + DW\,\,{\text{sin}}\,\varepsilon = 47.822$$

(23)

$$T = 2A \rho V_{i} \,\,V_{{\text{R}}} = {19}.{2497}$$

(24)

$${\text{Q }} = {\text{ A}} \rho V_{{\text{R}}} R^{2} \,\,\omega = {5}.{1341}\,{\text{Nm}}$$

(25)

$$P = Q \, \Omega = 6564.5629\,{\text{Nm}}$$

(26)

5 Mathematical Modeling and Autopilot Control System

The autopilot control system could be developed through modeling the necessary equations that need to be damped from the kinematics of flight [24].

The equations of motion for damping the pitching moments

$$\sum {\text{Pitching}}\,{\text{moments}} = \sum M_{{{\text{cg}}}} = I_{{y\ddot{\theta }}}$$

(27)

The pitching moment denoted by $M$ and the pitching angle denoted by $\theta$.

$M$ and $\theta$ in terms of the initial reference value are mentioned with subscript $0$, and the corresponding perturbation is mentioned by $\Delta$

$$M = M_{0} + \Delta M$$

(28)

$$\theta = \theta_{0} + \Delta \theta$$

(29)

If case that the reference moment which is denoted by $M_{0}$ becomes 0, then the Eq. (27) reduces to

$$\Delta M = I_{y} \Delta \ddot{\theta }$$

(30)

where

$$\Delta M = \frac{\partial M}{{\partial \alpha }} \Delta \alpha + \frac{\delta M}{{\delta \dot{\alpha }}}\Delta \dot{\alpha } + \frac{\partial M}{{\partial q}}\Delta q + \frac{\partial M}{{\partial \delta_{e} }} \Delta \delta_{e}$$

(31)

As there is a constrain applied to the C.G, the angle of attack will be identical to the pitch angle

$$\Delta \alpha = \Delta \theta$$

(32)

$$\Delta \dot{\theta } = \Delta \dot{\alpha }$$

(33)

$$\dot{\theta } = \Delta q$$

(34)

After substitution of the expression into Eq. (30), thereby rearranging would yield.

$$\Delta \ddot{\alpha } - \left( {M_{q} - M_{{\dot{\alpha }}} } \right)\Delta \dot{\alpha } + M_{\alpha } \Delta \alpha = M_{{\delta_{e} }} \Delta \delta_{e}$$

(35)

$$M_{q} = \frac{\partial M/\partial q}{{I_{y} }}$$

(36)

For the Plimp, the term $M_{\alpha }$ is negligible and could be eliminated in calculations.

Characteristics equation for Eq. (34) is

$$\lambda^{2} - \left( {M_{q} + M_{{\dot{\alpha }}} } \right)\lambda - M_{\alpha } = 0.$$

(37)

The undamped natural frequency $\omega_{n}$ of the system and damping ratio ζ can be determined by

$$\omega_{n} = \sqrt { - M_{\alpha } }$$

(38)

$$\zeta = - \frac{{\left( {M_{q} + M_{{\dot{\alpha }}} } \right)}}{{2\sqrt { - M_{\alpha } } }}$$

(39)

For a step change in rudder control, the solution to Eq. (35) would yield a damped sinusoidal motion, considering that the Plimp UAV has enough aerodynamic damping.

6 Results

We have obtained necessary parameters in hovering conditions, climbing conditions, and forward conditions are found out to be Table 5.

Table 5 Result table of various parameters

When we feed the rudder transfer function in the aircraft transfer function block in the damper block diagram and giving rudder servo equation as $\frac{10}{{\left( {S + 10} \right)}}$.

Washout circuit equation is given by $\frac{s}{{s + \frac{1}{\tau }}}$ and S (yrg) is given as $1.04\frac{v}{{{\raise0.7ex\hbox{${deg}$} \!\mathord{\left/ {\vphantom {{deg} {sec}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${sec}$}}}}$.

After obtaining the final equation through MATLAB, we use SISO tool toolbar to find the individual root locus and the final stability could be checked through the graph obtained and by varying the gain (Figs. 7, 8, 9, 10, 11, 12 and 13).

Fig. 7

Program for solving the transfer function MATLAB

Fig. 8

Graph of the solution for the transfer function in MATLAB

Fig. 9

Step response when gain k = 1, damping ratio = 0.203, natural frequency = 0.923, stable loop [8]

Fig. 10

Step response when gain k = 3, damping ratio = 0.036, natural frequency = 0.868, stable loop

Fig. 11

Step response when gain k = 3.3, damping ratio = 0, natural frequency = 0.865, stable loop

Fig. 12

Step response when gain k = 3.3, damping ratio = −0, natural frequency = 0.864, unstable loop

Fig. 13

Step response when gain k = 3.302651, damping ratio = −0, natural frequency = 0.864, neutrally stable loop

7 Summary

A detailed explanation of various design criteria of Plimp hybrid airship unmanned aerial vehicle has discussed along with their definitions and design constraints. Through the inputs of various stability parameters, necessary requirements of the hybrid VTOL Plimp airship have been defined in Sect. 3. The design of the Plimp unmanned aerial vehicle was modeled in CATIA software to get the exact geometric parameters. The calculations for performance parameters after weight estimation were performed in Sect. 4. The detailed calculations for different phases of flight were demonstrated. Further in Sect. 4, a detailed methodology of mathematical modeling of the stability parameters was derived and necessary equations that would be required for MATLAB code was obtained. In Sect. 5, the coding was performed in MATLAB SISO toolbox and through tuning various gain values, the designed system becomes stable as shown in the graphs of Sect. 6.

8 Conclusion

The designed tilt rotor stabilized Plimp unmanned aerial vehicle produce enough thrust, torque, and power with least possible drag. Also the MATLAB results show the stability for various gain values with different natural frequencies at various damping ratio. From the graph obtained in MATLAB SISO toolbox, we can see that the disturbances get damped and the system become stable. So this manuscript concludes that the modeled autopilot control system stabilizes the disturbances produced in Plimp hybrid airship UAV.