2.2.1 Single-point hanging model

The coordinate vector of the suspension point on the balloon in the body coordinate system is ${\bf{s}} = [0,0,d]^{\rm{T}}$ , where the superscript T represents transposition. With the cable length supposed as $l$ , at a certain equilibrium position, the angle between the cable and the plane ${o_s}{y_s}{z_s}$ is defined as the forward angle of inclination ${\theta _{\rm{L}}}$ , and the angle between the projection of the cable and the ${o_s}{z_s}$ axis of the plane ${o_s}{x_s}{z_s}$ is defined as the lateral angle of inclination ${\phi _{\rm{L}}}$ . The position of the pod in the suspension point coordinate system is ${\bf{l}} = {[{-}l\sin {\theta _{\rm{L}}},l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}},l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}]^{\rm{T}}}$ . The position vector of the pod in the body coordinate system ${\bf{p}} = {[{x_{\rm{p}}},\quad\!\!\! {y_{\rm{p}}},\quad\!\!\! {z_{\rm{p}}}]^{\rm{T}}}$ is described as

(1) \begin{align}{\bf{p}} = {\bf{l}} + {\bf{s}} = {[{-}l\sin {\theta _{\rm{L}}},\quad l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}},\quad l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}{\rm{ + d}}]^{\rm{T}}}\end{align}

Then, the velocity and acceleration of the pod relative to the body coordinate system are

(2) \begin{align}\dot{\bf{p}} = \left[ {\begin{array}{c@{\quad}c}{ - l\cos {\theta _{\rm{L}}}} {}&0\\[3pt]{ - l\sin {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}}& {}{l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}}\\[3pt]{ - l\sin {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}} {}&{ - l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}}\end{array}} \right]\left[ {\begin{array}{l}{{{\dot \theta }_{\rm{L}}}}\\[3pt]{{{\dot \phi }_{\rm{L}}}}\end{array}} \right] = {{\rm{A}}_{\rm{L}}}\left[ {\begin{array}{l}{{{\dot \theta }_{\rm{L}}}}\\[3pt]{{{\dot \phi }_{\rm{L}}}}\end{array}} \right]\end{align}

(3) \begin{align}\ddot{\bf{p }} & = \left[ {\begin{array}{c@{\quad}c}{ - l\cos {\theta _{\rm{L}}}} {}& 0\\[3pt]{ - l\sin {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}} {}&{l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}}\\[3pt]{ - l\sin {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}} {}&{ - l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}}\end{array}} \right]\left[ {\begin{array}{l}{{{\ddot \theta }_{\rm{L}}}}\\[8pt]{{{\ddot \phi }_{\rm{L}}}}\end{array}} \right]\nonumber\\[3pt] & + \left[ {\begin{array}{c}{l\sin {\theta _{\rm{L}}}\dot \theta {{_{\rm{L}}^{\rm{2}}}_{}}}\\[3pt]{ - l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}\dot \theta {{_{\rm{L}}^{\rm{2}}}_{}} - 2l\sin {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}{{\dot \phi }_{\rm{L}}}{{\dot \theta }_{\rm{L}}} - l\cos {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}\dot \phi _{\rm{L}}^2}\\[3pt]{ - l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}\dot \theta {{_{\rm{L}}^{\rm{2}}}_{}}{\rm{ + 2}}l\sin {\theta _{\rm{L}}}\sin {\phi _{\rm{L}}}{{\dot \phi }_{\rm{L}}}{{\dot \theta }_{\rm{L}}} - l\cos {\theta _{\rm{L}}}\cos {\phi _{\rm{L}}}\dot \phi _{\rm{L}}^2}\end{array}} \right] = {{\rm{A}}_{\rm{L}}}\left[ {\begin{array}{l}{{{\ddot \theta }_{\rm{L}}}}\\ {{{\ddot \phi }_{\rm{L}}}}\end{array}} \right] + {{\rm{B}}_{\rm{L}}}\end{align}

From Equation (3), because of the constraint with a constant cable length, the pod has two independent state variables: ${\theta _{\rm{L}}}$ and ${\phi _{\rm{L}}}$ .

2.2.2 Inertial motion of the pod in the body frame

When the body coordinate system has an instantaneous rotating angular velocity ${\boldsymbol{\omega }} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}p {}&q {}&r\end{array}} \right]^{\rm{T}}}$ , the position vector ${\bf{p}}$ produces an entrainment velocity ${\boldsymbol{\omega}} \times \bf{p}$ represented in the body coordinate system. The motion of the body coordinate system relative to the inertial coordinate system is expressed in the body coordinate system as ${{\bf{V}}_{\bf{O}}} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}u {}&v& {}w\end{array}} \right]^{\rm{T}}}$ , so the absolute velocity of the pod relative to the inertial coordinate system can be expressed in the body coordinate system as

(4) \begin{align}{{\bf{V}}_{\bf{P}}} = \left[ {\begin{array}{l}{{u_p}}\\[3pt]{{v_p}}\\[3pt]{{w_p}}\end{array}} \right] = {{\bf{V}}_{\bf{O}}} + \dot{\bf{p}} + \boldsymbol\omega \times \bf{p}\end{align}

The derivation of this equation gives the expression of the inertial acceleration of the pod in the body coordinate system as

(5) \begin{align}{{\bf{a}}_{\bf{P}}} & = {\dot{\bf{V}}}_{\bf{P}} = {\dot{\bf{V}}}_{\bf{O}} + {\dot{\boldsymbol{\omega}}} \times {\bf{p}} + {\boldsymbol{\omega}} \times {\dot{\bf{p}}} + {\boldsymbol{\omega}} \times ({{\bf{V}}_{\bf{O}}} + {\dot{\bf{p}}} + {\boldsymbol{\omega}} \times {\bf{p}}) + {\ddot{\mathbf{p}}}\nonumber\\[8pt]& = \left[ {\begin{array}{l}{\dot u}\\[3pt]{\dot v}\\[3pt]{\dot w}\end{array}} \right] + \left[ {\begin{array}{l}{\dot{p}}\\[3pt]{\dot q}\\[3pt]{\dot r}\end{array}} \right] \times \left[ {\begin{array}{l}{{x_{\rm{p}}}}\\[3pt]{{y_{\rm{p}}}}\\[3pt]{{z_{\rm{p}}}}\end{array}} \right] + \left[ {\begin{array}{l}p\\[3pt]q\\[3pt]r\end{array}} \right] \times \left( {\left[ {\begin{array}{l}u\\[3pt]v\\[3pt]w\end{array}} \right] + \left[ {\begin{array}{l}p\\[3pt]q\\[3pt]r\end{array}} \right] \times \left[ {\begin{array}{l}{{x_p}}\\[3pt]{{y_p}}\\[3pt]{{z_p}}\end{array}} \right]} \right) + 2\left[ {\begin{array}{l}p\\[3pt]q\\[3pt]r\end{array}} \right] \times \left[ {\begin{array}{l}{{{\dot x}_{\rm{p}}}}\\[3pt]{{{\dot y}_{\rm{p}}}}\\[3pt]{{{\dot z}_{\rm{p}}}}\end{array}} \right] + \left[ {\begin{array}{l}{{{\ddot x}_{\rm{p}}}}\\[3pt]{{{\ddot y}_{\rm{p}}}}\\[3pt]{{{\ddot z}_{\rm{p}}}}\end{array}} \right]\nonumber\\[8pt]& = \left[ {\begin{array}{l}{\dot u - vr + wq - {x_{\rm{p}}}({q^2} + {r^2}) + {y_{\rm{p}}}(pq - \dot r) + {z_{\rm{p}}}(pr + \dot q)}\\[3pt]{\dot v - wp + ur - {y_{\rm{p}}}({p^2} + {r^2}) + {z_{\rm{p}}}(qr - \dot{p}) + {x_{\rm{p}}}(pq + \dot r)}\\[3pt]{\dot w - uq + vp - {z_{\rm{p}}}({q^2} + {p^2}) + {x_{\rm{p}}}(rp - \dot q) + {y_{\rm{p}}}(rq + \dot{p})}\end{array}} \right] + 2\left[ {\begin{array}{l}{q{{\dot z}_{\rm{p}}} - r{{\dot y}_{\rm{p}}}}\\[3pt]{r{{\dot x}_{\rm{p}}} - p{{\dot z}_{\rm{p}}}}\\[3pt]{p{{\dot y}_{\rm{p}}} - q{{\dot x}_{\rm{p}}}}\end{array}} \right] + \left[ {\begin{array}{l}{{{\ddot x}_{\rm{p}}}}\\[3pt]{{{\ddot y}_{\rm{p}}}}\\[3pt]{{{\ddot z}_{\rm{p}}}}\end{array}} \right]\end{align}

where ${\dot{\bf{V}}_{\bf{O}}}+ {\ddot{\bf{p}}} + {\dot{\boldsymbol\omega}} \times {\bf{p}} + \boldsymbol\omega \times {\dot{\bf{p}}}$ is the inertial acceleration of the pod relative to the origin of the body coordinate system, and $\boldsymbol\omega \times ({\bf{V}}_{\bf{O}} + \dot{\bf{p}} + \boldsymbol\omega \times {\bf{p}})$ is the entrainment acceleration of the body coordinate system relative to the inertial coordinate system.

To simplify the model, this study considers the balloon and the pod to be in steady-state motion, ignoring the product form of the velocity vectors $\boldsymbol\omega \times {\dot{\bf{p}}}$ and ${\boldsymbol{\omega}} \times ({\bf{V}}_{\bf{O}}+ {\dot{\bf{p}}} + \boldsymbol\omega \times {\bf{p}})$ . Accordingly,

(6) \begin{align}{{\bf{a}}_{\bf{P}}} = \left[ {\begin{array}{c}{\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q}\\[3pt]{\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r}\\[3pt]{\dot w + - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p}}\end{array}} \right] + \left[ {\begin{array}{l}{{{\ddot x}_{\rm{p}}}}\\[3pt]{{{\ddot y}_{\rm{p}}}}\\[3pt]{{{\ddot z}_{\rm{p}}}}\end{array}} \right]\end{align}

2.2.3 External forces on the body frame

The inertial force on the pod is expressed in the body coordinate system as

(7) \begin{align}{{\bf{F}}_{{\bf{pI}}}} = {m_{\bf{P}}}{{\bf{a}}_{\bf{P}}} = \left\{ {\begin{array}{l}{{X_{pI}}}\\[3pt]{{Y_{pI}}}\\[3pt]{{Z_{pI}}}\end{array}} \right\}\end{align}

The gravity of the pod is decomposed into the body coordinate system as

(8) \begin{align}{{\bf{F}}_{{\bf{pG}}}} = \left\{ {\begin{array}{l}{{X_{pG}}}\\[3pt]{{Y_{pG}}}\\[3pt]{{Z_{pG}}}\end{array}} \right\} = \left\{ {\begin{array}{l}{ - {m_{\rm{p}}}g\sin \theta }\\[3pt]{{m_{\rm{p}}}g\sin \phi cos\theta }\\[3pt]{{m_{\rm{p}}}g\cos \phi \cos \theta }\end{array}} \right\}\end{align}

The balloon is actively controlled by the buoyancy adjustment system in the vertical direction and floats with the wind in the horizontal direction, so its aerodynamic force is manifested as drag in the vertical direction and as thrust in the horizontal direction. The aerodynamic force on the pod expressed in the body coordinate system is [Reference Carlson and Horn20]

(9) \begin{align}{{\bf{F}}_{{\bf{pA}}}} = \left\{ {\begin{array}{l}{{X_{pA}}}\\[3pt]{{Y_{pA}}}\\[3pt]{{Z_{pA}}}\end{array}} \right\} = \frac{1}{2}\rho S_p C_D(\bf{V}_{\bf{w}} - \bf{V}_{\bf{p}})|\bf{V}_{\bf{w}} - \bf{V}_{\bf{p}}|\end{align}

where $\rho $ is the air density, ${C_{\rm{D}}} \gt 0$ is the aerodynamic coefficient of the pod, ${S_p}$ is the reference area of the pod, and ${{\bf{V}}_{\bf{w}}} = {[\begin{array}{l}{{u_w}}\quad\!\! {}{{v_w}}\quad\!\! {{w_w}}\end{array}]^{\rm{T}}}$ is the wind velocity vector in the body coordinate system.

Under the body coordinate system, the cable force on the pod is ${{\bf{F}}_{{\bf{pL}}}}$ , which satisfies the force balance on the pod, as shown in Fig. 2.

(10) \begin{align}{{\bf{F}}_{{\bf{pA}}}}{\bf{ + }}{{\bf{F}}_{{\bf{pG}}}}{\bf{ + }}{{\bf{F}}_{{\bf{pL}}}} = {{\bf{F}}_{{\bf{pI}}}}\end{align}

Figure 2.

Diagram of the force balance on the pod.

From Equation (10), the pull force ${{\bf{F}}_{{\bf{pL}}}}$ represented in the body coordinate system is

(11) \begin{align}{{\bf{F}}_{{\bf{pL}}}} & = \left[ {\begin{array}{c}{{X_{pL}}}\\[3pt]{{Y_{pL}}}\\[3pt]{{Z_{pL}}}\end{array}} \right] = {m_p}\left[ {\begin{array}{c}{\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}}\\[3pt]{\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}}\\[3pt]{\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}}\end{array}} \right] - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}\left[ {\begin{array}{c}{({u_w} - {u_p})\left| {{u_w} - {u_p}} \right|}\\[3pt]{({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|}\\[3pt]{({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|}\end{array}} \right] - {m_{\rm{p}}}g\left\{ {\begin{array}{c}{ - \sin \theta }\\[3pt]{\sin \phi cos\theta }\\[3pt]{\cos \phi \cos \theta }\end{array}} \right\}\nonumber\\[9pt]& = \left\{ \begin{array}{l}{m_p}\left(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}\right) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}\left({u_w} - {u_p}\right)\left| {{u_w} - {u_p}} \right| + {m_{\rm{p}}}g\sin \theta \\[10pt]{m_p}\left(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}\right) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}\left({v_w} - {v_p}\right)\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta \\[10pt]{m_p}\left(\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}\right) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}\left({w_w} - {w_p}\right)\left| {{w_p} - {w_w}} \right| - {m_{\rm{p}}}g\cos \phi \cos \theta \end{array} \right.\end{align}

Meanwhile, the cable pull force ${{\bf{F}}_{{\bf{pL}}}}$ can be decomposed into the body coordinate system as

(12) \begin{align}{{\bf{F}}_{{\bf{pL}}}} = \left[ {\begin{array}{c}{{X_{pL}}}\\[3pt]{{Y_{pL}}}\\[3pt]{{Z_{pL}}}\end{array}} \right] = {\left\| {{{\bf{F}}_{{\bf{pL}}}}} \right\|_{\rm{2}}}\left[ {\begin{array}{c}{\sin {\theta _{\rm{L}}}}\\[3pt]{ - \sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}}\\[3pt]{ - \cos {\phi _{\rm{L}}}\cos {\theta _{\rm{L}}}}\end{array}} \right]\end{align}

Equation (12) indicates that two independent equations correspond to two independent state variables, as shown in Equation (3). Through the force balance equation of the pod, we have

(13) \begin{align}\left\{ \begin{array}{l}{X_{pL}}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} + {Y_{pL}}\sin {\theta _{\rm{L}}} = 0\\[3pt]{Y_{pL}} = {Z_{pL}}{\rm{tg}}{\phi _{\rm{L}}}\end{array} \right.\end{align}

Equation (11) is substituted into Equation (13), and the mass point motion equation of the pod is obtained as

(14) \begin{align}\left\{ \begin{array}{l}\left[{m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta \right]\\[10pt] = {m_p}(\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| - {m_{\rm{p}}}g\cos \phi \cos \theta ]{\rm{tg}}{\phi _{\rm{L}}}\\[10pt]\,\left[{m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right| + {m_{\rm{p}}}g\sin \theta \right]\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}\\[10pt] = - \left[{m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) - \frac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta \right]\sin {\theta _{\rm{L}}}\end{array} \right.\end{align}

2.3.1 Net-lift model in the whole process

Net-lift is defined here as the lift generated from the helium balloon, which is used to balance the weight of the structure. According to the ideal gas equation, the following relationship exists for the helium balloon, ballonet and external reference air:

(15) \begin{align}{V_{He}} &= \frac{{{m_{He}}{R_{He}}{T_{He}}}}{{{P_{He}}}}\nonumber\\[3pt]{\rho _{He}} &= \frac{{{P_{He}}}}{{{R_{He}}{T_{He}}}}\end{align}

(16) \begin{align}{V_{Air}} &= \frac{{{m_{Air}}{R_{Air}}{T_{Air}}}}{{{P_{Air}}}}\nonumber\\[3pt]{\rho _{Air}} &= \frac{{{P_{Air}}}}{{{R_{Air}}{T_{Air}}}}\end{align}

(17 \begin{align}\rho = \frac{P}{{{R_{Air}}T}}\end{align}

Here, ${R_{Air}} = 286.{\rm{7}}J/kg/K$ and ${R_{He}} = 20{\rm{78}}.{\rm{6}}J/kg/K$ are the specific gas constants of air and helium, respectively. ${m_{Air}}$ and ${m_{He}}$ are the internal air mass and helium mass, respectively. ${\rho _{Air}}$ and ${\rho _{He}}$ are the internal air density and helium density, respectively; ${T_{He}}$ and ${T_{Air}}$ are the helium temperature and internal air temperature, respectively; ${P_{He}}$ and ${P_{Air}}$ are the helium balloon pressure and internal air ballonet pressure, respectively; ${V_{He}}$ and ${V_{Air}}$ are the helium balloon volume and ballonet volume, respectively. The total volume of the buoyancy body is obtained as $V = {V_{Air}} + {V_{He}}$ . Given the density $\rho $ , pressure $P$ and temperature T of the external reference atmosphere, the net-lift from the buoyancy of the helium balloon is modelled as [Reference Carlson and Horn20]

(18) \begin{align}{L_{He}} & = g\rho {V_{He}} - {m_{He}}g + g\rho {V_{Air}} - {m_{Air}}g\nonumber\\[6pt]& = {m_{He}}g\left( {\frac{{\rho {V_{He}}}}{{{m_{He}}}} - 1} \right) + {m_{Air}}g\left( {\frac{{\rho {V_{Air}}}}{{{m_{Air}}}} - 1} \right)\nonumber\\[6pt]& = {m_{He}}g\left( {\frac{{\frac{{P{M_{Air}}}}{{RT}}\frac{{{m_{He}}R{T_{He}}}}{{{P_{He}}{M_{He}}}}}}{{{m_{He}}}} - 1} \right) + {m_{Air}}g\left( {\frac{{\frac{{P{M_{Air}}}}{{RT}}\frac{{{m_{Air}}R{T_{Air}}}}{{{P_{Air}}{M_{Air}}}}}}{{{m_{Air}}}} - 1} \right)\\[6pt]& = {m_{He}}g\left( {\frac{{{M_{Air}}}}{{{M_{He}}}}\frac{P}{{{P_{He}}}}\frac{{{T_{He}}}}{T} - 1} \right) + {m_{Air}}g\left( {\frac{P}{{{P_{Air}}}}\frac{{{T_{Air}}}}{T} - 1} \right)\nonumber\\[6pt]& = {m_{He}}g\left( {\frac{{{M_{Air}}}}{{{M_{He}}}}\frac{P}{{P + ({P_{He}} - P)}}\frac{{T + ({T_{He}} - T)}}{T} - 1} \right) + {m_{Air}}g\left( {\frac{P}{{P + ({P_{Air}} - P)}}\frac{{T + ({T_{Air}} - T)}}{T} - 1} \right)\nonumber \end{align}

Here, the universal gas constant $R = 8.314J/K/mol$ . ${M_{Air}}$ and ${M_{He}}$ are the molar masses of air and helium, respectively. $\delta {T_{He}} = {T_{He}} - T$ and $\delta {T_{Air}} = {T_{Air}} - T$ are the temperature difference between the inside and outside of the helium balloon and ballonet, respectively. $\delta {P_{He}} = {P_{He}} - P$ and $\delta {P_{Air}} = {P_{Air}} - P$ stand for the pressure difference between the inside and outside of the helium balloon and ballonet, respectively. In general, $\frac{{{P_{He}} - P}}{P}$ and $\frac{{{P_{Air}} - P}}{P}$ are small at low altitude, but they cannot be ignored when the balloon is approaching the climb ceiling.

For zero-pressure balloons [Reference Schallenkamp, Siewert and Lachenmeier21], no overpressure occurs, so net-lift is simplified as

(19) \begin{align}{L_{He}} = {m_{He}}g\left(\frac{{{M_{Air}}}}{{{M_{He}}}}\frac{{T + ({T_{He}} - T)}}{T} - 1\right) + {m_{Air}}g\left(\frac{{T + ({T_{Air}} - T)}}{T} - 1\right)\end{align}

Here, $\frac{{{T_{He}} - T}}{T}$ and $\frac{{{T_{Air}} - T}}{T}$ depend on the thermal characteristics of the balloon, as will be described in the following section.

For balloon long-time flying at a stable altitude, $\delta {T_{He}}$ and $\delta {T_{Air}}$ are neglected, then the net-lift becomes

(20) \begin{align}{L_{He}} = {m_{He}}g\left(\frac{{{M_{Air}}}}{{{M_{He}}}}\frac{P}{{P + ({P_{He}} - P)}} - 1\right) + {m_{Air}}g\left(\frac{P}{{P + ({P_{Air}} - P)}} - 1\right)\end{align}

2.3.2 Thermodynamic models during ascending and descending

During the ascending process, the balloon is not full of gas. Thus, the pressure of the helium balloon is [Reference Carlson and Horn20]

(21) \begin{align}{P_{He}} & = P = g\rho z\nonumber\\[3pt] \Rightarrow \frac{{d{P_{He}}}}{{dz}} & = \frac{{dP}}{{dz}} = - g\rho = - g\frac{{P{M_{Air}}}}{{RT}}\\[3pt] \Rightarrow \frac{{d{P_{He}}}}{{{P_{He}}dz}} & = \frac{{dP}}{{Pdz}} = - g\frac{{{M_{Air}}}}{{RT}}\nonumber\end{align}

where ${M_{Air}}$ is the molar mass of air. The thermal environment of the high-altitude balloon includes the internal and external environments. The heat transfer between the high-altitude balloon and external air involves radiation and the convection amongst the film, internal gases and external air. In consideration of the semitransparent property of the film, the thermal radiation amongst the film, external reference air and internal gases includes direct solar irradiation, reflected and scattered solar radiation and infrared radiation. Given that the film is very thin, its heat conduction is neglected for simplification. In accordance with the first law of thermodynamics, energy conservation equations for the helium balloon and the internal air ballonet are established as follows [Reference Shi, Song and Yao19]:

(22) \begin{align}{m_{He}}{C_{p,He}}\frac{{d{T_{He}}}}{{dt}} & = \sum {{Q_{He,i}} + {V_{He}}} \frac{{d{P_{He}}}}{{dt}}\nonumber\\[6pt]{m_{Air}}{C_{p,Air}}\frac{{d{T_{Air}}}}{{dt}} & = \sum {{Q_{Air,i}} + {V_{Air}}} \frac{{d{P_{Air}}}}{{dt}}(ascending)\\[6pt]{m_{Air}}{C_{p,Air}}\frac{{d{T_{Air}}}}{{dt}} & = \sum {{Q_{Air,i}} + {V_{Air}}} \frac{{d{P_{Air}}}}{{dt}} + {C_{p,Air}}({T_{Air}} - T)\frac{{d{m_{Air}}}}{{dt}}(descending)\nonumber\end{align}

where ${C_{p,He}}$ and ${C_{p,air}}$ are the specific heat of helium and air at constant pressure, respectively. $\sum {{Q_{He,i}}} $ and $\sum {{Q_{air,i}}} $ represent the heat transfer rate of helium and internal air with the environment, respectively. Equation (22) represents the energy conservation of helium and internal air. Specifically, the second equation is for the ascending phase, and the third one is for the descending phase, which considers the added heat from the pumped external air. The pump inflates the balloon by doing work, thereby generating heat. When deflating, the blower is driven by the internal and external pressure difference, and the energy required is negligible.

The heat transfer rate considered in Equation (22) includes the solar radiation heat $Ia{A_1}$ , the convective heat transferred between internal air and reference air ${K_1}{A_2}\left( {T - \left( {{T_{He}}{V_{He}} + {T_{Air}}{V_{Air}}} \right)/V} \right)$ and the heat transferred between internal air and helium $K_2^{}{A_3}\left( {{T_{He}} - {T_{Air}}} \right)$ , as shown below:

(23) \begin{align}\sum {{Q_{He,i}}} & = ({K_1}{A_2}\left( {T - \left( {{T_{He}}{V_{He}} + {T_{Air}}{V_{Air}}} \right)/V} \right) + Ia{A_1}){V_{He}}/V + K_2^{}{A_3}\left( {{T_{Air}} - {T_{He}}} \right)\nonumber\\[3pt]\sum {{Q_{Air,i}}} & = ({K_1}{A_2}\left( {T - \left( {{T_{He}}{V_{He}} + {T_{Air}}{V_{Air}}} \right)/V} \right) + Ia{A_1}){V_{Air}}/V + K_2^{}{A_3}\left( {{T_{He}} - {T_{Air}}} \right)\end{align}

Here, $I = 1367W/{m^2}$ is the estimated solar constant, $a = 0.4$ is the estimated efficiency of absorbing radiation, and ${A_1} = {V^{2/3}}$ is the estimated effective area of absorbing radiation. $K_1^{} = 30W/({m^2}K)$ and ${A_2} = {V^{2/3}}$ are the estimated heat transfer coefficient and the heat transfer area between the balloon and the environment, respectively. $K_2^{} = 10W/({m^2}K)$ and ${A_3} = V_{Air}^{2/3}$ are the estimated heat transfer coefficient and the heat transfer area between internal air and helium, respectively.

From Equations (15) and (21), we have

(24) \begin{align}{V_{He}}\frac{{d{P_{He}}}}{{dt}} & = {m_{He}}{R_{He}}{T_{He}}\frac{{d{P_{He}}}}{{{P_{He}}dz}}\frac{{dz}}{{dt}}\nonumber\\[3pt]& = - g{m_{He}}{R_{He}}{T_{He}}\frac{{{M_{Air}}}}{{{R_{Air}}T}}\dot z\\[3pt]& = - g{m_{He}}\frac{{{R_{He}}{T_{He}}}}{{{R_{Air}}T}}\dot z\nonumber\end{align}

Accordingly, we obtain

(25) \begin{align}{m_{He}}{C_{p,He}}\frac{{d{T_{He}}}}{{dt}} = \sum {{Q_{He,i}} - g{m_{He}}\frac{{{R_{He}}{T_{he}}}}{{RT}}\dot z} \end{align}

The current temperature of helium with vertical motion can be calculated using Equation (25), and the total net-lift can be determined via Equation (19).

2.3.3 Overpressure models with fixed volume ratio

When the balloon reaches the maximum altitude of 20km, a pressure difference $\delta {P_{Air}}$ occurs between the inside and outside of the balloon. Thus, the balloon has a fixed shape, and its volume is at the maximum value. This condition is defined as the steady state of the balloon. The pressure differences of the helium balloon and the ballonet are assumed to be equal, i.e. $\delta {P_{Air}} = \delta {P_{He}}$ , under the steady state for a given station-keeping altitude, which means that the shape will not change during the height adjustment process. The temperature difference between inside and outside the sphere for a long time at one altitude is ignored. For the mass conservation, we have

(26) \begin{align}{m_{Air}} &= \rho {V_{Air}}\left(1 + \frac{{\delta {P_{Air}}}}{P}\right)\nonumber\\[3pt]{m_{He}} &= {\rho _{He{\rm{0}}}}{V_{He}}\left(1 + \frac{{\delta {P_{He}}}}{P}\right)\\[3pt]V &= {V_{Air}} + {V_{He}}\nonumber\end{align}

where ${\rho _{He{\rm{0}}}}$ is the helium density under a normal pressure. For the balloon in a steady state, the volume ratio $k$ of the helium balloon to the ballonet is fixed and can be deduced as

(27) \begin{align}k &= \frac{{{V_{He}}}}{{{V_{Air}}}} = \frac{{{m_{He}}{\rho _{Air{\rm{0}}}}}}{{{m_{Air}}{\rho _{He{\rm{0}}}}}}\nonumber\\[3pt]{V_{Air}} &= V/(1 + k)\\[3pt]{V_{He}} &= V - {V_{Air}}\nonumber\end{align}

For a stable balance at any altitude, the balloon has a vertical balance, i.e.

(28) \begin{align}\left\{ \begin{array}{l}{\left. {g[(\rho - {\rho _{He{\rm{0}}}}){V_{He}} - {\rho _{He{\rm{0}}}}\dfrac{{\delta {\rho _{He}}}}{P}{V_{He}} - \rho \dfrac{{\delta {P_{Air}}}}{P}{V_{Air}}]} \right|_h} - ({m_{struc}} + {m_p})g = 0\\[8pt]{V_{He}} + {V_{Air}} = V\end{array} \right.\end{align}

where ${m_{stru}}$ is the mechanical structure mass, mainly including the two-way blower and the envelope.

The net-lift generated in Equation (28) is another form of Equation (20). If we substitute Equation (26) into Equation (28), the following relationship exists: $\rho = {\rho _{He{\rm{0}}}}\frac{{{M_{Air}}}}{{{M_{He}}}}$ .

2.4.1 Nonlinear model

From Equation (11), the force and moment generated by the cable pull force on the balloon expressed in the body coordinate system are

(29) \begin{align}\left\{ \begin{array}{l} - {{\bf{F}}_{{\bf{pL}}}} = \left\{ \begin{array}{l} - {m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| - {m_{\rm{p}}}g\sin \theta \\[6pt] - {m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_w} - {v_p}} \right| + {m_{\rm{p}}}g\sin \phi cos\theta \\[6pt] - {m_p}(\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_w} - {w_p}} \right| + {m_{\rm{p}}}g\cos \phi \cos \theta \end{array} \right.\\[30pt] - {\bf{s}} \times {{\bf{F}}_{{\bf{pL}}}} = \left\{ {\begin{array}{c}{ - d[{-}{m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_w} - {v_p}} \right| + {m_{\rm{p}}}g\sin \phi cos\theta ]}\\[6pt]{d[{-}{m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| - {m_{\rm{p}}}g\sin \theta ]}\\[6pt]0\end{array}} \right.\end{array} \right.\end{align}

where $\phi $ , $\theta $ and $\psi $ are the Euler angles of the body in the world coordinate system.

The dynamics of the balloon considering the cable pull force is

(30) \begin{align}\left\{ \begin{array}{l}{X_a} - mg\sin \theta - {X_{pL}} = m(\dot u - vr + wq)\\[3pt]{Y_a} + mg\cos \theta \sin \phi - {Y_{pL}} = m(\dot v - wp + ur)\\[3pt]{Z_a} + mg\cos \theta \cos \phi - {Z_{pL}} - {\rm{B}} = m(\dot w - uq + vp)\\[3pt]{L_a} + d{Y_{pL}} = {I_x}\dot{p} + ({I_z} - {I_y})rq\\[3pt]{M_a} - d{X_{pL}} = {I_y}\dot q + ({I_x} - {I_z})rp\\[3pt]{N_a} = {I_z}\dot r\end{array} \right.\end{align}

where the total mass of the balloon $m = {m_{Air}} + {m_{He}} + {m_{stru}} + {m_{add}}$ ; here, ${m_{add}}$ is the additional mass [Reference Carlson and Horn20]. ${I_x}$ , ${I_y}$ and ${I_z}$ are the inertial moments, $\rho $ is the density of the reference air, $Q$ is the dynamic pressure, and ${\rm{B}} = \rho V$ is the total buoyancy. ${X_a} = \frac{1}{2}\rho S{C_X}({u_w} - u)\left| {{u_w} - u} \right|$ , ${Y_a} = \frac{1}{2}\rho S{C_Y}({v_w} - v)\left| {{v_w} - v} \right|$ and ${Z_a} = \frac{1}{2}\rho S{C_Z}({w_w} - w)\left| {{w_w} - w} \right|$ are the aerodynamic components of the balloon along the three axes of the body coordinate system. ${L_a} = QS{l_{ref}}{C_l}$ , ${M_a} = QS{l_{ref}}{C_m}$ and ${N_a} = QS{l_{ref}}{C_n}$ are the aerodynamic moment components of the three axes around the body coordinate system. ${C_X},{C_Y},{C_Z},{C_l},{C_m}$ and ${C_n}$ are the aerodynamic coefficients of the balloon, $S$ is the reference area, and ${l_{ref}}$ is the reference length of the balloon.

Equations (14) and (30) constitute the whole 8-DOF system dynamics, i.e.

(31) \begin{align}\left\{ \begin{array}{l}[{m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta ]\\[3pt] = \,[{m_p}(\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| - {m_{\rm{p}}}g\cos \phi \cos \theta ]{\rm{tg}}{\phi _{\rm{L}}}\\[3pt]\,[{m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right| + {m_{\rm{p}}}g\sin \theta ]\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}\\[3pt] = - [{m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta ]\sin {\theta _{\rm{L}}}\\[3pt]m(\dot u - vr + wq) = {X_a} - mg\sin \theta - {m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| - {m_{\rm{p}}}g\sin \theta \\[3pt]m(\dot v - wp + ur) = {Y_a} + mg\cos \theta \sin \phi - {m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\\\qquad +\, {m_{\rm{p}}}g\sin \phi cos\theta \\[3pt]m(\dot w - uq + vp) = {Z_a} + mg\cos \theta \cos \phi - {m_p}(\dot w - {x_{\rm{p}}}\dot q + {y_{\rm{p}}}\dot{p} + {{\ddot z}_{\rm{p}}}) + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|\\\qquad +\, {m_{\rm{p}}}g\cos \phi \cos \theta - {\rm{B}}\\[3pt]{I_x}\dot{p} + ({I_z} - {I_y})rq = {L_a} + d({m_p}(\dot v - {z_{\rm{p}}}\dot{p} + {x_{\rm{p}}}\dot r + {{\ddot y}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta )\\[3pt]{I_y}\dot q + ({I_x} - {I_z})rp = {M_a} - d({m_p}(\dot u - {y_{\rm{p}}}\dot r + {z_{\rm{p}}}\dot q + {{\ddot x}_{\rm{p}}}) - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| + {m_{\rm{p}}}g\sin \theta )\\[3pt]{I_z}\dot r = {N_a}\end{array} \right.\end{align}

Equation (31) can be rearranged into

(32) \begin{align}\left\{ \begin{array}{l}{m_p}\dot v - {m_p}{\rm{tg}}{\phi _{\rm{L}}}\dot w - ({m_p}{z_{\rm{p}}} + {m_p}{\rm{tg}}{\phi _{\rm{L}}}{y_{\rm{p}}})\dot{p} + {m_p}{\rm{tg}}{\phi _{\rm{L}}}{x_{\rm{p}}}\dot q + {m_p}{x_{\rm{p}}}\dot r + {m_p}{{\ddot y}_{\rm{p}}} - {m_p}{\rm{tg}}{\phi _{\rm{L}}}{{\ddot z}_{\rm{p}}}\\[3pt] = \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| + {m_{\rm{p}}}g\sin \phi cos\theta - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|{\rm{tg}}{\phi _{\rm{L}}}\\[3pt] \qquad - {m_{\rm{p}}}g\cos \phi \cos \theta {\rm{tg}}{\phi _{\rm{L}}}\\[3pt]{m_p}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}\dot u - {m_p}\sin {\theta _{\rm{L}}}\dot v - {m_p}{z_{\rm{p}}}\sin {\theta _{\rm{L}}}\dot{p} - ({m_p}{y_{\rm{p}}}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} - {m_p}{x_{\rm{p}}}\sin {\theta _{\rm{L}}})\dot r + {m_p}{z_{\rm{p}}}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}\dot q\\[3pt] \qquad +\, {m_p}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}{{\ddot x}_{\rm{p}}} + {m_p}\sin {\theta _{\rm{L}}}{{\ddot y}_{\rm{p}}}\\[3pt] = \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right|\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} - {m_{\rm{p}}}g\sin \theta \sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\sin {\theta _{\rm{L}}}\\[3pt] \qquad +\, {m_{\rm{p}}}g\sin \phi cos\theta \sin {\theta _{\rm{L}}}\\[1pt](m + {m_p})\dot u + {m_p}{z_{\rm{p}}}\dot q - {m_p}{y_{\rm{p}}}\dot r + {m_p}{{\ddot x}_{\rm{p}}} = {X_a} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| - (mg + {m_{\rm{p}}}g - B)\sin \theta\\[3pt] \qquad +\, mvr - mwq\\[1pt](m + {m_p})\dot v - {m_p}{z_{\rm{p}}}\dot{p} + {m_p}{x_{\rm{p}}}\dot r + {m_p}{{\ddot y}_{\rm{p}}} = {Y_a} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| + (mg + {m_{\rm{p}}}g - B)\sin \phi cos\theta\\[3pt] \qquad +\, mwp - mur\\[1pt](m + {m_p})\dot w + {m_p}{y_{\rm{p}}}\dot{p} - {m_p}{x_{\rm{p}}}\dot q + {m_p}{{\ddot z}_{\rm{p}}} = {Z_a} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|\\[3pt] \qquad +\, (mg + {m_{\rm{p}}}g - B)\cos \phi \cos \theta muq - mvp\\[1pt] - {m_p}d\dot v + ({I_x} + {m_p}d{z_{\rm{p}}})\dot{p} - {m_p}d{x_{\rm{p}}}\dot r - {m_p}d{{\ddot y}_{\rm{p}}} = {L_a} - d\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - (mg{z_g}\\[3pt] \qquad +\, (d + {z_p}){m_{\rm{p}}}g)\sin \phi cos\theta - ({I_z} - {I_y})rq\\[1pt]{m_p}d\dot u + ({I_y} + {m_p}d{z_{\rm{p}}})\dot q - {m_p}d{y_{\rm{p}}}\dot r + {m_p}d{{\ddot x}_{\rm{p}}} = {M_a} + d\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_w} - {u_p}} \right| - (mg{z_g}\\[3pt] \qquad +\, (d + {z_p}){m_{\rm{p}}}g)\sin \theta - ({I_x} - {I_z})rp\\[3pt]{I_z}\dot r = {N_a}\end{array} \right.\end{align}

2.4.2 Point mass model

The balloon is assumed to be stable in level flight, such that the movement of the pod is balanced. The changes in the angular velocity and acceleration of the balloon are ignored; thus, $p = q = r = 0$ , and $\dot{p} = \dot q = \dot r = 0$ . The point mass model of the platform is simplified to

(33) \begin{align}\left\{ \begin{array}{l}{m_p}\dot v + {m_p}{{\ddot y}_{\rm{p}}} - {m_p}{\rm{tg}}{\phi _{\rm{L}}}\dot w - {m_p}{\rm{tg}}{\phi _{\rm{L}}}{{\ddot z}_{\rm{p}}} = - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|{\rm{tg}}{\phi _{\rm{L}}} - {m_{\rm{p}}}g{\rm{tg}}{\phi _{\rm{L}}}\\[3pt]\qquad +\, \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\\[10pt]{m_p}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}\dot u + {m_p}\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}}{{\ddot x}_{\rm{p}}} + {m_p}\sin {\theta _{\rm{L}}}\dot v + {m_p}\sin {\theta _{\rm{L}}}{{\ddot y}_{\rm{p}}}\\[3pt] = \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right|\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\sin {\theta _{\rm{L}}}\\[7pt]m\dot u + {m_p}(\dot u + {{\ddot x}_{\rm{p}}}) = \dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{X}}}({u_w} - u)\left| {u - {u_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right| - mg\sin \theta - {m_{\rm{p}}}g\sin \theta \\[7pt]m\dot v + {m_p}(\dot v + {{\ddot y}_{\rm{p}}}) = \dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{Y}}}({v_w} - v)\left| {v - {v_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| + mg\cos \theta \sin \phi\\[7pt]\qquad +\, {m_{\rm{p}}}g\sin \phi cos\theta \\[1pt]m\dot w + {m_p}(\dot w + {{\ddot z}_{\rm{p}}}) = \dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{Z}}}({w_w} - w)\left| {w - {w_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| + mg\cos \theta \cos \phi\\[7pt]\qquad +\, {m_{\rm{p}}}g\cos \phi \cos \theta - {\rm{B}}\end{array} \right.\end{align}

Equation (3) is combined with Equation (33) to form the kinetic equation of the mass point system.

2.4.3 Static equilibrium

This study assumes that the platform is stable in level flight and that the pod is statically balanced; hence, $p = q = r = 0$ , $\dot{p} = \dot q = \dot r = 0$ , $\dot u = \dot v = \dot w = 0$ and ${\ddot x_{\rm{p}}} = {\ddot y_{\rm{p}}} = {\ddot z_{\rm{p}}} = 0$ hold. Then, the cable pull force is simplified to

(34) \begin{align}{{\bf{F}}_{{\bf{pL}}}} = \left[ \begin{array}{l} - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right| + {m_{\rm{p}}}g\sin \theta \\[10pt] - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g\sin \phi cos\theta \\[10pt] - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| - {m_{\rm{p}}}g\cos \phi \cos \theta \end{array} \right]\end{align}

The platform is in a static equilibrium state, so

(35) \begin{align}\left\{ {\begin{array}{l}{\dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{X}}}({u_w} - u)\left| {u - {u_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right| = 0}\\[10pt]{\dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{Y}}}({v_w} - v)\left| {v - {v_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| = 0}\\[10pt]{\dfrac{{\rm{1}}}{{\rm{2}}}\rho S{C_{\rm{Z}}}({w_w} - w)\left| {w - {w_w}} \right| + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| + mg + {m_{\rm{p}}}g - {\rm{B = 0}}}\\[10pt]{ - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right|{\rm{tg}}{\phi _{\rm{L}}} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| - {m_{\rm{p}}}g{\rm{tg}}{\phi _{\rm{L}}}}\\[10pt]{\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - u)\left| {u - {u_w}} \right|\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\sin {\theta _{\rm{L}}}}\end{array}} \right.\end{align}

Supposing that the attitude of the helium balloon is negligible, i.e. $\theta = \phi = 0$ , this study further simplifies the cable pull force to

(36) \begin{align}{{\bf{F}}_{{\bf{pL}}}} = \left[ \begin{array}{l} - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - {u_p})\left| {{u_p} - {u_w}} \right|\\[10pt] - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\\[10pt] - \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| - {m_{\rm{p}}}g\end{array} \right]\end{align}

When the moving speed of the pod and the wind speed are equal, the horizontal component of the cable pull force of the platform is zero. The vertical component of the cable pull force is the magnitude of the pod weight. Hence, the pod static balance is simplified to

(37) \begin{align}\left\{ \begin{array}{l}\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right| = [\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| + {m_{\rm{p}}}g]{\rm{tg}}{\phi _{\rm{L}}}\\[10pt]\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({u_w} - u)\left| {u - {u_w}} \right|\sin {\phi _{\rm{L}}}cos{\theta _{\rm{L}}} + \dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|\sin {\theta _{\rm{L}}} = 0\end{array} \right.\end{align}

Under this condition, the hanging angle of the pod satisfies the following equation:

(38) \begin{align}\left\{ \begin{array}{l}{\phi _{\rm{L}}} = a{\rm{tg}}\left( {\dfrac{{\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|}}{{\dfrac{1}{2}\rho {S_p}{C_{\rm{D}}}({w_w} - {w_p})\left| {{w_p} - {w_w}} \right| + {m_{\rm{p}}}g}}} \right)\begin{array}{l}{}\end{array}or\begin{array}{l}{}\end{array}{\phi _{\rm{L}}} = a{\rm{tg}}\left( {\dfrac{{{Y_{pL}}}}{{{Z_{pL}}}}} \right)\\[22pt]{\theta _{\rm{L}}} = a{\rm{tg}}\left( {\dfrac{{ - ({u_w} - u)\left| {u - {u_w}} \right|\sin {\phi _{\rm{L}}}}}{{({v_w} - {v_p})\left| {{v_p} - {v_w}} \right|}}} \right)\begin{array}{l}{}\end{array}or\begin{array}{l}{}\end{array}{\theta _{\rm{L}}} = a{\rm{tg}}\left( {\dfrac{{ - {X_{pL}}}}{{\sqrt {{Y_{pL}}^{\rm{2}} + {Z_{pL}}^{\rm{2}}} }}} \right)\end{array} \right.\end{align}

From Equation (38), the lateral angle of inclination ${\phi _{\rm{L}}}$ is determined by the ratio of the lateral force to the vertical component force experienced by the pod. The forward angle of inclination ${\theta _{\rm{L}}}$ is determined by the ratio of the forward force experienced by the pod to the resultant force in the plane ${o_s}{x_s}{y_s}{z_s}$ .

Table 1.

Parameters of the platform