Design of Flyby Trajectories with Powered Gravity and Aerogravity Assist Maneuvers

Abstract

In this paper, we investigate flyby trajectories combining powered gravity assist (PGA) with aerogravity assist (AGA) in the planar elliptic restricted three-body problem (PERTBP). The patched flyby trajectory is divided into three portions: the PGA, AGA and ballistic portions, successively. In the PGA portion, continuous thrusts are conducted to change the speed and drive the altitude of a vehicle below the atmosphere edge. A simple flight-path angle guidance algorithm for three stages is used to design the orbit in the AGA portion. Taking the Sun–Mars PERTBP system as an example, flyby trajectories around Mars combining PGA with AGA are constructed and discussed in detail. In addition, numerical results show that the elliptical effect of the model should not be ignored, and it is necessary to investigate patched flyby orbits in the PERTBP.

1. Introduction

Gravity assist (GA) is a classical technique to change the path and energy of a vehicle through a spacecraft flyby or swing-by near a celestial body. GA could improve the heliocentric velocity of the vehicle by using the gravity of the planet, and reduce the fuel consumption and flight time of the mission to a certain extent. It is of great significance for the development of detection missions to achieve long-distance and multi-scientific goals. Furthermore, GA has been widely used in deep space exploration missions that have been implemented and planned in the future, such as the “Mariner” Mercury exploration mission, the “Voyager” exoplanet exploration mission, the “Cassini” Saturn exploration mission, the Galileo–Jupiter orbiter, the “Europa Clipper” exploration mission, and so forth.

Furthermore, some improved GA techniques were proposed to increase variations in the velocity, energy and direction of a vehicle during the flyby. For example, Prado [1] presented a concept named powered swing-by (PSB) or powered gravity-assist (PGA). That is, the vehicle performs one or more impulse maneuvers during its closest approach to the central planet to improve the energy variation efficiency. After that, the most convenient position and direction of the velocity impulse by a parametric analysis of a one-impulse PSB were found by Casalino et al. [2]. Orbital characters of one-impulse PGA were analyzed in the circular restricted three-body problem (CRTBP) [3] and the elliptic restricted three-body problem (ERTBP) [4]. Qi and de Ruiter [5] developed a method for calculating the energy variation for PGAs with continuous thrusts in the ERTBP. However, the major drawback of the PGA is the propellant cost.

Some researchers proposed another concept named aerogravity assist (AGA) to improve the effect of flyby through aerodynamic forces. It uses the planetary atmosphere to increase the flyby time of a vehicle to obtain a greater path change. Miele and Wang [6] proposed a flight-path angle guidance scheme (or Gamma Guidance) for the AGA to realize the coplanar, aero-assisted orbital transfer from high Earth orbit to low Earth orbit. McRonald and Randolph [7] analyzed AGA maneuvers of the waverider through the Mars atmosphere. Lohar et al. [8] studied the optimal AGA trajectory under the heat constraint, and found that the heating rate constraint is a significant limitation in the optimization. Johnson and Longuski [9] applied a graphical method to identify potential AGA trajectories. Based on this method, they constructed a transfer orbit to Pluto using a Venus–Mars–Venus series of AGAs. Pessina et al. [10] implemented a preliminary analysis of a wide range of mission opportunities, offered by either AGA or GA manoeuvres. Based on their results, AGA can allow faster missions than simple GA at comparable $\Delta V$-cost. Mazzaracchio [11] proposed a three-stage guidance law in the AGA orbit based on the Gamma guidance of Miele and Wang [6], and studied the hyperbolic trajectory to Venus performed by a waverider. Some scholars [12,13,14] applied the AGA to achieve orbit insertion around a planet by using the aerodynamic drag generated by the atmosphere of the planet to decelerate. Furthermore, Piñeros et al. [15] investigated the energy changes of a spacecraft that performs an impulse during the atmospheric flight around the Earth. They defined this flyby process as the powered aerogravity-assist (PAGA). Qi and de Ruiter [16] analyzed PAGA trajectories by assuming that continuous thrusts are conducted in atmospheric flight. Numerical results indicated that PAGA orbits can achieve an even larger velocity or energy change than AGA orbits. However, the major drawback of the AGA is that the vehicle has to be designed specifically to provide high lift and low drag at a hypersonic speed.

Compared with the PGA technique, an advantage of the AGA technique is that it does not cost any on-board fuel. However, it should be noted that the basic premise of occurrence for the AGA requires that the altitude of the periapsis of a vehicle with respect to the planet must be below its atmosphere edge, so there are higher requirements for the accuracy of the insert orbit. If the original GA orbit has a large deviation at the insert point and does not satisfy the above AGA requirements, continuous thrusts could be conducted before periapsis to reduce its altitude below the atmosphere edge. It should be noted that some researchers have used the deep space maneuvers (DSMs) before flyby to re-target the altitude of the periapsis of the flyby orbit. However, in this paper, we focus on the orbital motion near the flyby planet, where the PGA is adopted. Hence, for arbitrary original GA orbits, especially those not satisfying the AGA premise, an innovative technique combining the PGA and AGA can be adopted to efficiently change the orbital characters of a vehicle. In this paper, this technique, named PGA+AGA, is investigated. We assume that an entire PGA+AGA trajectory is divided into three portions: the PGA, AGA and ballistic portions. The PGA portion with continuous thrust will be constructed by the optimal control problem solver. A simple flight-path angle guidance algorithm is proposed to design the AGA portion. Different from the simple two-body model used for PAGA maneuvers in Ref. [16], a more accurate Sun–planetary ERTBP model is adopted in this paper to depict orbital characters during the planetary flyby. Furthermore, we will design different AGA portions by lift modulation, and the results of PGA+AGA orbits for different periapsis epochs are analyzed and compared.

The structure of this paper is organized into five parts. In Section 2, we introduce background materials for this paper, such as the ERTBP and PGA. In Section 3, the AGA portion is designed by lift modulation for AGA maneuvers and analyzed. In Section 4, the results of PGA+AGA orbits are discussed and compared. Section 5 is the conclusion of this paper.

2. Background

In this section, the background of this paper, including the dynamical model and the PGA portion, is introduced.

2.1. Dynamical Model

In the solar system, none of the orbital eccentricities of celestial bodies are exactly equal to zero, i.e., their orbits are not circular, and some of them are even highly eccentric. For example, the Sun–Mars system has an eccentricity of 0.0935; the Sun–Mercury system has an eccentricity of 0.2056; the Sun–Pluto system has an eccentricity of 0.2488. Therefore, the elliptic-restricted three-body problem (ERTBP), a more precise model than the circular restricted three-body problem (CRTBP), was proposed to depict the motion of a test particle, such as a spacecraft, comet or asteroid, in the gravitational field of two primary bodies moving in elliptic motions around their barycenter [17]. Based on Ref. [18], the eccentricity can also be added in the patched conic model. However, the patched conic model cannot depict the detailed process of the flyby orbit, such as the energy and angular momentum. According to Ref. [19], in the ERTBP, the mechanism of the GA can be directly revealed without simplification, and orbital parameters, like the energy and angular momentum, can be obtained easily.

In this paper, the Sun–planet planar ERTBP (PERTBP) is used as the dynamical model of flyby orbits. The PERTBP is a classical model to describe the planar motion of a vehicle in the gravitational field of two primaries, which move in elliptic orbits about their barycenter O (see Figure 1). $m_1$ and $m_2$ denote the masses of the Sun and planet, respectively. A vehicle passes through the planetary atmosphere, and its dimensionless velocity vector in the rotating frame is denoted by v. The $XY$ coordinates are inertial coordinates, whose X axis is along the vector from O to the periapsis of the elliptic orbit of the planet. The $xy$ coordinates are pulsating rotating coordinates, whose origin O is similar to that of the inertial coordinates with the x axis pointing along the vector from the Sun to the planet. For the PERTBP system, the eccentricity of the elliptic orbit is represented by e, and the mass ratio $\mu$ is equal to $m_2/(m_1+m_2)$. According to Ref. [17], the true anomaly f of $m_2$ in the elliptic orbit is used as the independent variable instead of time t, and the pulsating distance between the Sun and planet, i.e., r, is used as the length unit. Then, the equation of motion of a flyby orbit with external forces in the dimensionless pulsating rotating coordinates is expressed as [17]

$$\begin{array}{cc}\hfill x^{\prime \prime }& =2y^{\prime }+{\overline{u}}_x+T_x,\hfill \\ \hfill y^{\prime \prime }& =-2x^{\prime }+{\overline{u}}_y+T_y,\hfill \end{array}$$

(1)

where the superscript ${\left(\right)}^{\prime }$ denotes differentiation with respect to f. $T={(T_x,T_y)}^T$ is the dimensionless external force on a vehicle except for gravitational forces, for example, in this paper, the propulsive force $T_{pga}$ and the aerodynamic force $T_{aga}$. $\overline{u}=\overline{U}/(1+ecosf)$ is the effective potential in the ERTBP; ${\overline{u}}_x$ and ${\overline{u}}_y$ are the partial derivatives of $\overline{u}$ with respect to the position variables x and y, with

$$\overline{U}=\frac{1}{2}(x^2+y^2)+\frac{1-\mu }{r_1}+\frac{\mu }{r_2}+\frac{1}{2}\mu (1-\mu ),$$

while

$$\begin{array}{cc}\hfill r_1& =\sqrt{{(x+\mu )}^2+y^2},\hfill \\ \hfill r_2& =\sqrt{{(x+\mu -1)}^2+y^2}.\hfill \end{array}$$

The dimensionless velocity vector in the rotating frame is denoted by $v={(x^{\prime },y^{\prime })}^T$. The length unit r is time-varying and is expressed as,

$$r\left(f\right)=\frac{a(1-e^2)}{1+ecosf}=\frac{p}{g},$$

(2)

where

$$g=1+ecosf.$$

(3)

a and $p=a(1-e^2)$ are the semimajor axis and semi-latus rectum of the orbital motion of $m_2$ around the barycenter, respectively.

The true anomaly f and the time t have the following relationship [19],

$$\begin{array}{cc}\hfill \frac{\mathrm{d}f}{\mathrm{d}t}& =\sqrt{\frac{G(m_1+m_2)}{a^3{(1-e^2)}^3}}{(1+ecosf)}^2\hfill \\ & =\sqrt{G(m_1+m_2)}\frac{g^2}{\sqrt{p^3}}\hfill \end{array}$$

(4)

where G is the Newtonian gravitational constant.

The inertial energy of the spacecraft with respect to the barycenter O of the PERTBP system is defined as

$$E=K+U,$$

(5)

where K and U are the kinetic energy and the potential energy of the spacecraft with respect to the barycenter, respectively. As given in Ref. [19],

$$\begin{array}{cc}\hfill K& =\frac{G(m_1+m_2)}{a}\frac{g^2}{2(1-e^2)}\left[{\left(x^{\prime }-y+Ax\right)}^2+{\left(y^{\prime }+x+Ay\right)}^2\right],\hfill \\ \hfill U& =-\frac{G(m_1+m_2)}{a}\frac{g}{1-e^2}\left(\frac{1-\mu }{r_1}+\frac{\mu }{r_2}\right),\hfill \end{array}$$

where $A=esinf/(1+ecosf)$.

The inertial velocity vector of a vehicle with respect to the barycenter O of the PERTBP system is given by

$$V_o=r\left(f\right)M\left(f\right)\frac{\mathrm{d}f}{\mathrm{d}t}\left(\begin{array}{c}{x}^{\prime }-y+Ax\\ y^{\prime }+x+Ay\end{array}\right),$$

(6)

where

$$M=\left(\begin{array}{cc}cosf& -sinf\\ sinf& cosf\end{array}\right).$$

The relative velocity vector of a vehicle with respect to $m_2$ is given by

$$V_2=r\left(f\right)M\left(f\right)\frac{\mathrm{d}f}{\mathrm{d}t}{v}_2,$$

(7)

where

$$v_2=\left(\begin{array}{c}{x}^{\prime }-y+A(x-1+\mu )\\ y^{\prime }+(x-1+\mu )+Ay\end{array}\right).$$

(8)

Figure 2 shows a schematic of a patched flyby trajectory combining the powered gravity assist (PGA) and aerogravity assist (AGA), i.e., PGA+AGA, in a circular neighborhood of the planet. The bold red line denotes the PGA+AGA orbit, and the bold dashed line denotes the incoming orbit without the propulsive and aerodynamic forces (or the original GA orbit). The atmosphere edge is the sensible range of the atmosphere. Outside the atmosphere edge, the density of the atmosphere is so small that the effects of the atmosphere on the orbit are negligible. As mentioned before, a PGA+AGA trajectory is divided into three portions: the PGA, AGA and ballistic portions. The PGA portion is the orbit section between points $P_1$ and $P_2$, which are the entering points of the circular neighborhood and atmosphere edge, respectively. In this portion, continuous thrusts are working to drive the altitude of a vehicle below the atmosphere edge. The AGA portion is the orbit section in the atmosphere, where point $P_3$ denotes the departing point of the atmosphere. In this portion, we assume that the thrust engine shuts down. The ballistic portion is the orbit section between points $P_3$ and $P_4$, where point $P_4$ denotes the departing point of the circular neighborhood. In this portion, neither the thrust engine nor aerodynamic forces work, and a vehicle is only affected by gravitational forces from the Sun and planet. The PGA and AGA portions are the emphases of this paper because they depend on our design method.

In this paper, some variation indices of a PGA+AGA orbit through a circular neighborhood of the planet are used, including the velocity variation, energy variation and total turn angle. These variation indices can be computed by the difference between the indices at points $P_4$ and $P_1$. For example, let $V_{o1}$ and $V_{o4}$ denote the inertial velocities with respect to the barycenter O at points $P_1$ and $P_4$, respectively, then the velocity variation is given by $\Delta V=\parallel V_{o4}-V_{o1}\parallel$. Let $E_1$ and $E_4$ denote the inertial energies at points $P_1$ and $P_4$, respectively, then the energy variation is given by $\Delta E=E_4-E_1$. Let $V_{21}$ and $V_{24}$ denote the inertial velocities with respect to $m_2$ at points $P_1$ and $P_4$, respectively, then the total turn angle is given by $\delta =arccos\left[(V_{21}·V_{41})/\left(\parallel V_{21}\parallel \parallel V_{41}\parallel \right)\right]$.

As shown in Qi and de Ruiter [20], the parameter set of the original GA orbit is denoted by ${\mathsf{\Omega }}_c=\left\{f_0,\sigma ,{\psi }_0,e_c,D\right\}$. The incoming orbit of a vehicle is approximated by a hyperbolic orbit with respect to the planet. $f_0$ denotes the value of true anomaly f of the planet in the elliptic orbit when a vehicle is located at the periapsis of the incoming orbit with respect to the planet. As stated in Ref. [19], since f is a slow variable with respect to the true anomaly of a vehicle $\theta$ (see Figure 1), we treat $f_0$ as a constant in the PGA+AGA problem. $\sigma$ is the index of the motion direction of the incoming orbit with respect to the planet: prograde (anticlockwise incoming orbit with respect to the planet) and retrograde (clockwise incoming orbit with respect to the planet) motions correspond to 1 and −1, respectively. It should be noted that in this paper, only prograde incoming orbits are used, but the design method of a PGA+AGA orbit is available for both prograde and retrograde cases. ${\psi }_0$ represents the phase angle of its periapsis with respect to the x axis (see Figure 2). $e_c$ is the eccentricity of the incoming orbit with respect to the planet, which is required to be larger than 1 in this paper, i.e., the incoming orbit should be a hyperbolic orbit with respect to the planet. D is the periapsis distance of the incoming orbit from the center of the planet.

2.2. PGA Portion

In this section, assuming that there is no aerodynamic force in the atmosphere, the PGA portion with continuous thrust is designed to reach the given target periapsis altitude. Based on Ref. [5], in the PGA portion, the dimensionless propulsive force of the PERTBP can be expressed as,

$$T_{pga}={\mathrm{TU}}^2\frac{T_c{(1-e^2)}^2}{a{(1+ecosf)}^3}\left(cos\alpha v_t+sin\mathbf{\alpha }{v}_{\perp }\right),$$

(9)

where $T_c$ is the magnitude of the thrust acceleration and its unit is m/s². $\mathrm{TU}$ is the time unit of the PERTBP system, and is equal to $\sqrt{a^3/\left[G(m_1+m_2)\right]}$. $\alpha$ represents the thrust direction, and is defined as the turn angle of the maneuver with respect to the velocity vector of the incoming orbit, and is measured in an anticlockwise manner (see Figure 2). $v_t$ and $v_{\perp }$ are a pair of orthogonal basis vectors, and are defined as

$$\begin{array}{cc}\hfill v_t& =v/\parallel v\parallel ={\left(x^{\prime },y^{\prime }\right)}^T/\sqrt{x^{\prime 2}+y^{\prime 2}},\hfill \\ \hfill v_{\perp }& ={\left(-y^{\prime },x^{\prime }\right)}^T/\sqrt{x^{\prime 2}+y^{\prime 2}}.\hfill \end{array}$$

(10)

Based on Equation (9), the magnitude of the thrust $T_c$ and the thrust direction $\alpha$ are two control variables in the PGA portion. We assume that $T_c\in [0,T_{\max }]$, where $T_{\max }$ is the maximum magnitude of the available thrust acceleration. According to the experiences in Ref. [5], we use the continuous optimal control problem solver GPOPS-II (next-generation optimal control software) to design the PGA portion.

Based on Ref. [5], the main role of the continuous thrust before periapsis in the optimal PGA process is to reduce the periapsis distance to increase the variation of velocity or energy, while the main role of the thrust after periapsis in the optimal PGA process is to directly increase the variation in velocity or energy. In the PGA+AGA issue analyzed in this paper, the AGA portion plays a dominant role in the change of velocity or energy. Since the PGA is implemented before the AGA portion, according to the above analysis, its role is set as a re-targeting of the altitude to take full advantage of the effect of AGA. Correspondingly, in this paper, the cost function of the optimal control problem is

$$J=H\left(f_p\right)-H_{\mathrm{tar}},$$

(11)

where $H_{\mathrm{tar}}$ is the given target altitude of periapsis with respect to the planet. $f_p$ is the value of f when a vehicle is located at periapsis with respect to the planet. It should be noted that the value of $f_p$ is usually different from $f_0$ mentioned before, because $f_0$ denotes the time of periapsis of the incoming orbit (see the dashed line in Figure 2) with respect to the planet, but $f_p$ is the time of periapsis of the real orbit (see the red solid line in Figure 2) with respect to the planet. $H\left(f\right)$ is the altitude of a vehicle at f, and can be expressed as

$$H\left(f\right)=ar\left(f\right)\sqrt{{\left[x\left(f\right)-1+\mu \right]}^2+y{\left(f\right)}^2}-R_p.$$

(12)

where $R_p$ is the radius of the planet.

In the PGA portion, the dynamic constraints can be obtained by Equations (1) and (9); however, when a vehicle enters the planetary atmosphere, based on our above assumption, we require that both $T_{pga}$ and $T_{aga}$ are zero. The path constraint is

$$R-R_p⩾H\left(f\right)⩾H_{\mathrm{tar}},$$

(13)

where R is the radius of the circular neighborhood of the planet. In this paper, we require that R is equal to $R_{\mathrm{SOI}}/2$, where $R_{\mathrm{SOI}}$ is the radius of the sphere of influence (SOI) of the planet and can be obtained from $R_{\mathrm{soi}}=a{\left[\mu /(1-\mu )\right]}^{2/5}$ [21].

The boundary conditions include the initial condition at point $P_1$, which is fixed once the incoming orbit is given, and the terminal condition at periapsis. Let $\gamma$ denote the flight-path angle of a vehicle, and it can be obtained from

$$\gamma =arccos\frac{V_2·R_2}{V_2{R}_2},$$

(14)

where

$$R_2=r\left(f\right)M\left(f\right)\left(\begin{array}{c}x-1+\mu \\ y\end{array}\right),$$

(15)

is the position vector of a vehicle with respect to the planet in the inertial frame. $V_2=\parallel V_2\parallel$ and $R_2=\parallel R_2\parallel$. The terminal condition requires

$$\gamma \left(f_p\right)=0.$$

(16)

In the implementation of the optimization, we use the original GA orbit as the initial guess. Based on the above equations, we can obtain the PGA portion using the solver GPOPS-II.

3. AGA Portion

3.1. Flight-Path Angle Guidance Algorithm

In this section, we adopt the flight-path angle guidance algorithm similar to Qi and de Ruiter [16] to design the AGA portion. Since the PERTBP model we used in this paper is different from the two-body model of Ref. [16], the flight-path angle guidance algorithm of Ref. [16] should be changed to fit the new model.

Figure 3 shows a schematic of the AGA portion, including the descent, level-flight and ascent phases. The end points of the descent and level-flight phases are denoted by $P_d$ and $P_a$, respectively.

We take the same assumptions as Ref. [11] for the AGA portion. Readers can refer to this reference for details. In addition, we assume that the atmospheric density $\rho$ is modeled as strictly exponential and defined by

$$\rho ={\rho }_{0}exp\left(-\beta H\right),$$

(17)

where $\beta$ denotes the inverse of the constant scale altitude of the atmosphere, and ${\rho }_0$ is the reference density on the surface. It should be noted that, in practice, atmospheric density uncertainties are major current challenges for AGA maneuvers, but those uncertainties were ignored in this paper. The aerodynamic force can be expressed as,

$$T_{aga}=T_l+T_d,$$

(18)

where $T_l$ and $T_d$ are the lift and drag forces in the dimensionless rotating frame, respectively. They can be obtained by

$$\begin{array}{cc}\hfill T_l& =-\frac{500\sigma r\rho s{C}_L}{m_s}∥v_2∥v_1,\hfill \\ \hfill T_d& =-\frac{500r\rho s{C}_D}{m_s}∥v_2∥v_2.\hfill \end{array}$$

(19)

where $v_1$ is a vector perpendicular to $v_2$, and is defined as

$$\begin{array}{c}\hfill v_1=\sigma \left[\begin{array}{c}-y^{\prime }-(x+\mu -1)-Ay\\ x^{\prime }-y+A(x+\mu -1)\end{array}\right].\end{array}$$

(20)

Based on the assumption of the drag polar,

$$C_D=C_{D0}+K{C}_L^2,$$

(21)

where

$$\begin{array}{cc}\hfill & C_{D0}=C_L^{*}/\left(2E^{*}\right),\hfill \\ \hfill & K=C_{D0}/{\left(C_L^{*}\right)}^2,\hfill \end{array}$$

(22)

where $E^{*}$ is the maximum lift-to-drag ratio, and $C_L^{*}$ is the lift coefficient at $E^{*}$.

In the design process of the AGA portion, a vehicle is controlled by the lift coefficient $C_L$. Normally, $C_L$ is positively valued with the bank angle determining the orientation of the lift vector. However, since the bank angle is excluded in our assumption, $C_L$ is allowed to take on both position and negative values [22]. We assume that $C_L\in [-C_{L\max },C_{L\max }]$, where $C_{L\max }$ is the allowable maximum value of $C_L$. The flight-path angle guidance algorithm is differentiated for the three phases.

Let ${\tilde{C}}_L$ denote the value of $C_L$ for the level flight. As given in Ref. [11],

$$V_2\dot{\gamma }=\frac{\rho s{C}_L{V}_2^2}{2m_s}-\left(\frac{GM\mu }{R_2^2}-\frac{V_2^2}{R_2}\right)cos\gamma .$$

(23)

where the flight-path angle $\gamma$ can be obtained from Equation (14). Let $\dot{\gamma }$ and $\gamma$ both be equal to zeros, then we can obtain the condition of the level fight [11]:

$${\tilde{C}}_L=\frac{2m_s}{\rho s}\left(\frac{GM\mu }{R_2^2{V}_2^2}-\frac{1}{R_2}\right).$$

(24)

At the starting point of the descent phase, $P_1$, the lift force L is assumed to provide a positive lift force, so $C_L$ should be larger or equal to 0. Based on Ref. [16], it can be set as the maximum value, $C_{L\max }$. At the end point of the descent phase, $C_L$ should obey the condition of the level flight, so $C_L={\tilde{C}}_L$. Let ${\gamma }_1$ denote the initial flight-path angle of a vehicle at point $P_1$. When the flight-path angle $\gamma$ increases from ${\gamma }_1$ to 0, $C_L$ in the descent phase is assumed to vary from $C_{L\max }$ to ${\tilde{C}}_L$ linearly with $\gamma$. According to [16], a coefficient $k_{cld}$ is introduced as a multiplier to adjust the variation of $C_L$. The feedback control form for the descent phase can be expressed as [16]:

$$C_L=k_{cld}\left[{\tilde{C}}_L+\left(C_{L\max }-{\tilde{C}}_L\right)\frac{\gamma }{{\gamma }_1}\right],$$

(25)

When the magnitude of $\left|\gamma \right|$ is smaller than a small critical value ${\epsilon }_{\gamma }$, we postulate that the descent phase finishes and the level-flight phase begins. In this paper, ${\epsilon }_{\gamma }$ is set as 0.005 deg. For the level-flight phase, as mentioned before, the altitude reduction of a vehicle is quite slow, which can be regarded as constant, i.e., $\gamma \approx 0$, so we obtain the following control form,

$$C_L={\tilde{C}}_L,$$

(26)

$\Delta t_l$ is denoted as the duration of the level-flight phase. Based on Ref. [16], when the flight-path angle $\gamma$ increases from 0 to $-{\gamma }_1$, $C_L$ in the ascent phase is linearly changed from ${\tilde{C}}_L$ to $C_{L\max }$ with $\gamma$. The coefficient $k_{cla}$ is introduced as a multiplier to adjust the variation in $C_L$ in the ascent phase. For the ascent phase, the following feedback control form is expressed as [16]:

$$C_L=k_{cla}\left[{\tilde{C}}_L-\left(C_{L\max }-{\tilde{C}}_L\right)\frac{\gamma }{{\gamma }_1}\right],$$

(27)

where $\gamma$ and ${\tilde{C}}_L$ can be obtained from Equations (14) and (24), respectively.

If the initial state at point $P_1$ is given, the AGA portion is uniquely determined by coefficients $k_{cld}$, $k_{cla}$ and $\Delta t_l$. In addition, heat constraints play an important role in the AGA portion [11,23]. In this paper, the peak heat rate and the total heat load in the stagnation region of a vehicle are taken into consideration. Based on Ref. [24], the convective heating rate ${\dot{q}}_c$ along the atmospheric trajectory can be expressed as,

$${\dot{q}}_c=C_c{\left(\frac{\rho }{r_n}\right)}^{\frac{1}{2}}{V}^3,$$

(28)

where $r_n$ is the radius of the stagnation region, and $C_c$ is a constant dependent on the composition of the atmosphere.

According to Ref. [25], the radiative heating rate ${\dot{q}}_r$ can be written as,

$${\dot{q}}_r=C_r{r}_n^b{\rho }^{c}F\left(V\right),$$

(29)

where $F\left(V\right)$ is the tabulated value, and is determined by the flight velocity V and the atmospheric composition. The exponents b and c are functions of $\rho$ and V. $C_r$ is a coefficient relying on the planetary atmosphere and is regarded as a constant for a given planet.

Therefore, the total heat load is obtained from

$$Q={\int }_0^{\Delta t_a}\left({\dot{q}}_c+{\dot{q}}_r\right)\mathrm{d}t.$$

(30)

where $\Delta t_a$ is the duration of the AGA portion.

3.2. Comparison of AGA Portion

Mazzaracchio [11] proposed a flight-path angle guidance algorithm using lift modulation for AGA maneuvers on hyperbolic trajectories. Similar to our design process, the lift coefficient was tuned according to a three-stage guidance law, in which a linear flight-path angle guidance was assumed for both the ascending and descending atmospheric flight branches with a different gain coefficient in the feedback control form. Then, Mazzaracchio took a case study of a simple AGA maneuver to Venus as an application of his guidance scheme. In this subsection, we compare our AGA orbit with the results in Ref. [11]. For comparison, data of constants, vehicle properties and boundary values are the same as those in Ref. [11]. Since in this pure AGA case, the altitude of the periapsis of the original GA orbit is equal to the given target altitude of periapsis, 110 km, we assume that the thrust engine does not work in the PGA portion, i.e., $T_c=0$.

As stated in Section 3, we can adjust $k_{cld}$, $k_{cla}$ and $\Delta t_l$ to design the AGA portion. In this case, $k_{cld}$, $k_{cla}$ and $\Delta t_l$ are set as 1, 0.95 and 177 s, respectively. We use these values to simulate the trajectory course in Ref. [11]. In this way, we can compare their orbital changes of AGA maneuvers. Figure 4 shows the AGA orbit obtained from our method in the Sun–Venus dimensionless rotating frame. As can be seen from the figure, the AGA orbit and the original GA orbit separate after the Venus atmospheric fly-through, and the AGA orbit obtains a larger turn angle compared with the GA orbit. Figure 5 displays the parameter profiles in the atmosphere of Figure 4, where $V_2$ denotes the magnitude of the relative velocity with respect to Venus and can be computed by Equation (7). Comparing Figure 5 with Figures 3–6 in Ref. [11], we find that except for the $C_L$ curve, the change curves of H, $V_2$ and $\gamma$ versus time $t_{aga}$ in Figure 5 are quite similar to Mazzaracchio’s results. Data of our result and Mazzaracchio’s results are listed in

Table 1. Comparison with results in Ref. [11].

Method	${\mathit{V}}_{\mathit{\infty }}^{-}$ (km/s)	${\mathit{V}}_{\mathit{\infty }}^{+}$ (km/s)	$\mathbf{\Delta }\mathit{V}$ (km/s)	${\mathit{H}}_{min}$ (km)	${\mathit{V}}_{max}$ (km/s)	$\mathit{\delta }$ (deg)	$\mathbf{\Delta }{\mathit{t}}_{\mathit{a}}$ (s)
Ours	14	10.99	15.66	105.14	17.32	73.44	373.65
Ref. [11]	14	11.76	15.60	110.10	17.36	73.95	375.91

, where $V_{\infty }^{-}$ and $V_{\infty }^{+}$ represent the initial and final hyperbolic excess velocities with respect to Venus. $H_{\min }$ and $V_{\max }$ are the minimum of H and the maximum of $V_2$, respectively. From this table, we can observe that for the same given $V_{\infty }^{-}$, the values of our AGA orbit are quite similar to those in Ref. [11]. Since the trajectory profile in Ref. [11] correlated with the optimal trajectories presented in Ref. [8], based on the above comparisons, we conclude that our flight-path angle guidance algorithm also provides suboptimal AGA trajectories. Since the target periapsis altitude $H_{\mathrm{tar}}$ is set as 110 km, Mazzaracchio’s $H_{\min }$ is closer to $H_{\mathrm{tar}}$ than our result. Actually, in the feedback control form of Ref. [11], the lift coefficient $C_L$ is regulated by the term $(H-H_{\mathrm{tar}})$, so its $H_{\min }$ can reach $H_{\mathrm{tar}}$ more closely. However, our guidance law of the descent phase is only influenced by $\gamma$. Therefore, our algorithm cannot ensure that $H_{\min }$ is approximately equal to $H_{\mathrm{tar}}$. Compared with the guidance law in Ref. [11], our feedback control form is simpler, and coefficients of the control, $k_{cld}$, $k_{cla}$ and $\Delta t_l$, are easier to understand and change.

4. Results and Discussion

In this section, the results of PGA+AGA trajectories are demonstrated and discussed. As an application of our design method, PGA+AGA trajectories around Mars have been discussed. Data of constants, vehicle properties and boundary values are listed in

Table 2. Constants, vehicle properties and boundary values in the Mars case.

Name	Symbol	Value
Mass ratio of ERTBP	$\mu$	3.253253$\times {10}^{-7}$
Mass coefficient	$G(m_1+m_2)$	1.327128$\times {10}^{11}$ km3/s2
Semi-major axis of ERTBP	a	2.2792$\times {10}^8$ km
Eccentricity of ERTBP	e	0.0935
Radius of Mars	$R_m$	3396.2 km
Radius of SOI	$R_{\mathrm{SOI}}$	5.7914$\times {10}^5$ km
Reference density	${\rho }_0$	0.02 kg/m3
Inverse scale altitude	$\beta$	0.094 km−1
Sensible altitude of atmosphere	$H_{\mathrm{atm}}$	500 km
Vehicle mass	m	1500 kg
Aerodynamic reference area	S	30 m2
Maximum lift-to-drag ratio	$E^{*}$	3
Lift coefficient at maximum $L/D$	$C_L^{*}$	0.034
Sutton-Graves constant	$C_c$	1.9027$\times {10}^{-8}$ (${\mathrm{kg}}^{1/2}·$m)/cm2
Nose radius	$r_n$	1 m
Maximum $C_L$	$C_{L\max }$	0.7
Maximum thrust acceleration	$T_{\max }$	0.003 m/s2
Initial eccentricity	$e_c$	4.5
True anomaly of the periapsis	$f_0$	0 deg
Initial periapsis altitude	$H_{\mathrm{ini}}$	10,000 km
Target periapsis altitude	$H_{\mathrm{tar}}$	60 km

. The atmospheric data are adopted from data in Ref. [26]. Two cases of PGA+AGA orbits with ${\psi }_0={90}^{\circ }$ and ${270}^{\circ }$ are studied, respectively.

4.1. Case of ${\psi }_0={90}^{\circ }$

The case of ${\psi }_0={90}^{\circ }$ is first demonstrated. Figure 6 displays the control variables of the PGA portion obtained from the GPOPS-II solver. As we can see from Figure 6, $\alpha$ is distributed in a range of $[{93}^{\circ },{107}^{\circ }]$, so the direction of the continuous thrust is approximately perpendicular to the vehicle velocity in the PGA portion.

According to the previous discussion in Section 3, the AGA portion is determined by parameters $k_{cld}$, $k_{cla}$ and $\Delta t_l$. Once the descent and level-flight phases are specified by $k_{cld}$ and $\Delta t_l$, the ascent phase is only influenced by $k_{cla}$. In the numerical simulation of this subsection, since the vehicle velocity at the entering point of the atmosphere is relatively small, we should reduce the velocity loss to ensure that some PGA+AGA orbits can pass through the circular neighborhood of Mars. To this end, $k_{cla}$ is set as 0 to minimize the drag force. We adjust $k_{cld}$ and $\Delta t_l$ to construct the AGA portion. Let $e_p$ denote the instantaneous eccentricity of a vehicle with respect to Mars in the circular neighborhood. Figure 7 shows the orbital results of the AGA portions with different $k_{cld}$ and $\Delta t_l$, where ${\left(e_p\right)}_{\mathrm{end}}$ is the value of $e_p$ at the departing point $P_4$ and ${\left(V_2\right)}_{\mathrm{end}}$ is the value of the relative velocity $V_2$ at point $P_3$. From the figure, we can observe that with the increase in $\Delta t_l$, both ${\left(e_p\right)}_{\mathrm{end}}$ and ${\left(V_2\right)}_{\mathrm{end}}$ decrease. This result agrees with our expectations. Due to the drag force in the AGA portion, a larger flight time of the level-flight phase $\Delta t_l$ results in a larger velocity loss. In addition, we find that almost all $H_{\min }$s are smaller than the target altitude $H_{\mathrm{tar}}$, and they are mainly influenced by $k_{cld}$ rather than $\Delta t_l$. $H_{\min }$ decreases with the increase in $k_{cld}$.

Heat results are displayed in Figure 8, where ${\left({\dot{q}}_c\right)}_{\max }$ denotes the peak value or the maximum of ${\dot{q}}_c$ in the AGA portion. Similar to the result in Ref. [26], numerical results also indicate that the peak convective heating rate ${\dot{q}}_c$ is significantly larger than the peak radiative heating rate ${\dot{q}}_r$, so the distribution of the latter is not shown here. The total heat load Q mainly comes from the convective heat. From the figure, we find that ${\left({\dot{q}}_c\right)}_{\max }$ is mainly affected by $k_{cld}$; however, the influence of $\Delta t_l$ is quite small. Recalling the result of $H_{\min }$ in Figure 7, we conclude that ${\left({\dot{q}}_c\right)}_{\max }$ increases with the decrease in $H_{\min }$. This result accords with our expectation because, based on Equations (17) and (28), ${\dot{q}}_c$ is very sensitive to the altitude H. In addition, Q increases with the increase in $k_{cld}$ and $\Delta t_l$.

Based on the definition of $e_p$, a vehicle is captured by Mars when ${\left(e_p\right)}_{\mathrm{end}}<1$, while a vehicle flies by Mars when ${\left(e_p\right)}_{\mathrm{end}}>1$. Therefore, the ‘Capture’ and ‘Flyby’ regions could be easily found in the distribution of ${\left(e_p\right)}_{\mathrm{end}}$ in Figure 7. The ‘Capture’ region can be used in other related aerotechniques, such as aerocapture, atmospheric re-entry and so forth. In this paper, we focus on the PGA+AGA orbits in the ‘Flyby’ region. Figure 9 illustrates the results of PGA+AGA orbits in the ‘Flyby’ region. According to Ref. [19], when $f=0^{\circ }$ and ${\psi }_0={90}^{\circ }$, GA orbits have the maximum energy loss. As we can see from the figure, for a given $k_{cld}$, the energy loss increases monotonically with the decrease in $\Delta t_l$. However, the total turn angle $\delta$ increases monotonically with the increase in $\Delta t_l$. In addition, the maximum of the velocity variation $\Delta V$ is located in the interior rather than the border.

Based on Figure 9, three PGA+AGA solutions with the maximum velocity variation $\Delta V$, energy loss and turn angle can be easily obtained and named as PGA+AGA orbits 1∼3, respectively.

Table 3. Data of five orbits with ${\psi }_0={90}^{\circ }$.

Orbit	${\mathit{k}}_{\mathbf{cld}}$	$\mathbf{\Delta }{\mathit{t}}_{\mathit{l}}$ (s)	${\left({\mathit{e}}_{\mathit{p}}\right)}_{\mathbf{end}}$	$\mathbf{\Delta }\mathit{V}$ (km/s)	$\mathbf{\Delta }\mathit{E}$ (km2/s2)	$\mathit{\delta }$ (deg)
GA orbit	–	–	4.4994	1.5959	$-40.0741$	25.6384
PGA orbit	–	–	2.0426	3.4067	$-89.0183$	61.8849
PGA+AGA orbit 1	0.82	142	1.3412	3.9754	$-80.2733$	104.2773
PGA+AGA orbit 2	0.3	0	1.8183	3.6915	$-94.0209$	71.2620
PGA+AGA orbit 3	0.83	393	1.0081	2.7968	$-34.7093$	159.2501

lists data of PGA+AGA orbits 1∼3. For comparison, data of the original GA and PGA orbits are also shown in this table. The PGA orbit is obtained from the design method in Section 2.2, but its target altitude is the sensible altitude of the atmosphere $H_{\mathrm{atm}}$ (500 km) rather than 60 km. Therefore, there are no aerodynamic forces during the entire PGA orbit. Figure 10 displays the GA, PGA and PGA+AGA orbits 1∼3 in the Sun–Mars dimensionless rotating frame. Based on Table 3, compared with the GA and PGA orbits, the $\Delta V$ of the PGA+AGA orbit 1 rises 149.1% and 16.7%, respectively; the energy loss of the PGA+AGA orbit 2 rises by 134.6% and 5.6%, respectively; $\delta$ of PGA+AGA orbit 3 rises by 133.6 deg and 97.4 deg, respectively. Apparently, PGA+AGA orbits could efficiently improve the change in orbital elements.

Figure 11 shows the parameter profiles of five orbits with time t, where $E_p$ is the inertial energy with respect to Mars. As we can see from Figure 10, continuous thrusts in the PGA portion effectively decrease periapsis altitudes of the PGA orbit and PGA+AGA orbits 1∼3 compared to the original GA orbit. However, in Figure 11, we find that in the PGA portion, the differences among the GA, PGA orbits and PGA+AGA orbits 1∼3 are not noticeable in terms of E and $E_p$. This result can be explained by recalling the $\alpha$ curve in Figure 6. As mentioned before, the direction of the continuous thrust is approximately perpendicular to the vehicle velocity in the PGA portion. Hence, the work of the continuous thrust on a vehicle is quite small, and there is no obvious change in the E and $E_p$ profiles during the PGA portion. However, the continuous thrust could change the shape of the trajectory, so we find that the $e_p$ curves of the PGA orbit and PGA+AGA orbits 1∼3 decrease from 4.5 to about 2. Correspondingly, their periapsis altitudes are reduced. After the atmospheric fly-through, the E values of the GA and PGA orbits both decrease, and since the periapsis altitude of the PGA orbit is smaller, it can obtain a larger energy loss. However, the $E_p$ and $e_p$ values of the GA and PGA orbits both remain after the atmospheric fly-through, because no other external forces exist except gravitational forces. PGA+AGA orbits 1∼3 separate after the atmospheric fly-through due to different control coefficients $k_{cld}$ and $\Delta t_l$ (see Figure 10). Correspondingly, their parameter profiles are variant after the atmospheric fly-through (see Figure 11).

Figure 12 and Figure 13 illustrate the profiles of orbital and heat parameters in the AGA portions of PGA+AGA orbits 1∼3 in Figure 10. As we can see from Figure 12, for PGA+AGA orbits 1∼3, their maximums of $V_2$ all appear in the vicinity of $P_d$. In the level-flight and ascent phases, a vehicle progressively loses its speed due to drag. From Figure 13, we can observe that peak values of ${\dot{q}}_c$ also occur in the vicinity of $P_d$. This result is understandable because a vehicle has a larger V and a smaller H in the vicinity of $P_d$. In addition, from Figure 13, we can observe that PGA+AGA orbit 3 with a larger $\Delta t_l$ has a noticeably larger heat load Q.

4.2. Case of ${\psi }_0={270}^{\circ }$

Next, the PGA+AGA orbits with ${\psi }_0={270}^{\circ }$ are analyzed. Numerical results indicate that the $\alpha$ and $T_c$ curves of the case of ${\psi }_0={270}^{\circ }$ are similar to those in Figure 6, so they are not displayed here to avoid repetition. In the AGA portions, we also fix $k_{cla}=0$, and adjust $k_{cld}$ and $\Delta t_l$ to construct AGA orbits. Numerical results also show that the distributions of ${\left(e_p\right)}_{\mathrm{end}}$, ${\left(V_2\right)}_{\mathrm{end}}$, $H_{\min }$, ${\left({\dot{q}}_c\right)}_{max}$ and Q in the case of ${\psi }_0={270}^{\circ }$ are same as those in Figure 7 and Figure 8. Similarly, they are not displayed here to avoid repetition.

For the PGA+AGA orbits with ${\psi }_0={270}^{\circ }$, in the ‘Flyby’ region, the distributions of $\Delta V$, $\Delta E$ and $\delta$ are displayed in Figure 14. According to Ref. [19], when $f=0^{\circ }$ and ${\psi }_0={270}^{\circ }$, GA orbits have the maximum energy gain. From this figure, we can observe that for a given $k_{cld}$, the energy gain increases monotonically with the decrease in $\Delta t_l$. Similar to Figure 9, the value of $\delta$ increases monotonically with the increase in $\Delta t_l$. However, compared with Figure 9, $\Delta V$ increases monotonically with the increase in $\Delta t_l$, so its maximum occurs at the maximum of $\Delta t_l$, which is the same as the distribution of $\delta$.

Based on Figure 14, two PGA+AGA solutions with the maximum $\Delta V$ and energy gain are chosen and named as PGA+AGA orbits 1 and 2, respectively. According to Figure 14, PGA+AGA orbit 1 also has the maximum turn angle $\delta$.

Table 4. Data of four orbits with ${\psi }_0={270}^{\circ }$.

Orbit	${\mathit{k}}_{\mathbf{cld}}$	$\mathbf{\Delta }{\mathit{t}}_{\mathit{l}}$ (s)	${\left({\mathit{e}}_{\mathit{p}}\right)}_{\mathbf{end}}$	$\mathbf{\Delta }\mathit{V}$ (km/s)	$\mathbf{\Delta }\mathit{E}$ (km2/s2)	$\mathit{\delta }$ (deg)
GA orbit	–	–	4.4994	1.5959	40.0741	25.6384
PGA orbit	–	–	2.0426	3.7093	89.3518	61.8850
PGA+AGA orbit 1	0.9	386	1.0092	5.2370	23.6754	159.3214
PGA+AGA orbit 2	0.15	0	1.8571	4.0576	93.3228	69.3531

lists data of PGA+AGA orbits 1 and 2. For comparison, data of the original GA and PGA orbits are also shown in this table. The GA, PGA orbits and PGA+AGA orbits 1∼2 are shown in Figure 15 in the Sun–Mars dimensionless rotating frame. According to Table 4, compared with the GA and PGA orbits, $\Delta V$ of PGA+AGA orbit 1 rises by 228.2% and 41.2%, respectively; the energy gain of PGA+AGA orbit 2 rises by 132.9% and 4.4%, respectively; $\delta$ of PGA+AGA orbit 1 rises by 133.7 deg and 97.4 deg, respectively. Hence, we conclude that PGA+AGA orbits could efficiently increase the change in orbital elements.

Figure 16 shows the parameter profiles of four orbits with time. From Figure 15 and Figure 16, we find that some conclusions in the case of ${\psi }_0={90}^{\circ }$ are similar to those in the case of ${\psi }_0={270}^{\circ }$. For the PGA orbit and PGA+AGA orbits 1 and 2, the continuous thrust cannot efficiently change E and $E_p$ during the PGA portion due to its thrust direction, but the continuous thrust can significantly decrease $e_p$. After the atmospheric fly-through, the E curves of the GA and PGA orbits both decrease, and since the periapsis altitude of the PGA orbit is smaller, it can obtain a larger energy gain. The $E_p$ and $e_p$ curves of the GA and PGA orbits both remain after the atmospheric fly-through. PGA+AGA orbits 1 and 2 separate after the atmospheric fly-through due to different control coefficients $k_{cld}$ and $\Delta t_l$ (see Figure 15).

Figure 17 and Figure 18 display the profiles of orbital and heat parameters in the AGA portions of PGA+AGA orbits 1 and 2 in Figure 15. Similar to results in Figure 12 and Figure 13, we find that the maximums of both V and ${\dot{q}}_c$ occur in the vicinity of point $P_d$.

4.3. Influence of Elliptic Model

To the authors’ best knowledge, almost all previous works of the AGA orbits were investigated in the two-body model. In this paper, PGA+AGA orbits are constructed and analyzed in the PERTBP. To demonstrate the special character of this problem in the PERTBP, in this subsection, the influence of the true anomaly of the periapsis $f_0$ is discussed. We still take the Sun–Mars PERTBP as an example, and except for $f_0$, other data of constants, vehicle properties and boundary values in Table 2 are also adopted. In the numerical simulation, ${\psi }_0$ is set as ${90}^{\circ }$, and AGA coefficients $k_{cld}$, $k_{cla}$ and $\Delta t_l$ are set as 0.2, 0 and 100 s, respectively. Figure 19 shows the distributions of $\Delta V$ and $\Delta E$ with different $f_0$. The results of the GA, PGA and PGA+AGA orbits of the ERTBP are compared. In addition, if we assume that the eccentricity of the Mars orbit is zero, results of the Sun–Mars CRTBP can also be obtained using the above parameters. As we can see from the figure, $f_0$ has no influence on the results of the time-invariant CRTBP. Compared with the results of the CRTBP, $f_0$ has obvious influences on $\Delta V$ and $\Delta E$ for the GA, PGA and PGA+AGA orbits of the ERTBP, but its effects are different. Based on the amplitudes of the $\Delta V$ curves, we find that the effect of $f_0$ on $\Delta V$ of the GA and PGA orbit is smaller than that of PGA+AGA orbits. In addition, the minimum $\Delta V$ of GA orbits is located at aphelion (i.e., $f_0={180}^{\circ }$), but at this point, PGA and PGA+AGA orbits obtain their maximum $\Delta V$. According to the amplitudes of the $\Delta E$ curves with different $f_0$, we find that $f_0$ has more obvious influences on PGA and PGA+AGA orbits than GA orbits. Beyond that, we also observe that the addition of the continuous thrust and aerodynamic forces can change phase positions of the $\Delta E$ curves. As we can see, the maximum energy loss of GA orbits occurs at perihelion (i.e., $f_0=0^{\circ }$), which accords with the conclusion in Ref. [19]; however, numerical computation indicates that the maximum energy losses of PGA and PGA+AGA orbits are located at $f_0=341.5^{\circ }$ and $313.8^{\circ }$, respectively.

If we just focus on the difference between the Sun–Mars ERTBP and Mars-centric two-body problem at a specific epoch, the influence of the Sun on the short flyby process is slight and can be approximately ignored. However, an advantage of the ERTBP is that we could integrally analyze the influence of the orbital eccentricity of the Mars orbit on the flyby. As we can see from Figure 19, the fluctuation caused by $f_0$ reflected the effect of the ERTBP. On the other hand, in this figure, comparisons of curves between the PGA and PGA+AGA indicated influences of AGA on $\Delta V$ and $\Delta E$, while comparisons of curves between the GA and PGA indicated influences of PGA on $\Delta V$ and $\Delta E$. From this figure, we find that the effect of the PGA is most significant, and effects of the AGA and ERTBP model have the same magnitude on $\Delta V$. Based on the above results, we conclude that the elliptical effect of the model should not be ignored, and it is necessary to investigate the PGA+AGA problem in the PERTBP.

In addition, it should be noted that the eccentricity effect is not the only perturbation of the more precise model compared to the simple CRTBP. Some researchers studied perturbations from other gravities. For example, Koon et al. [27], Lantoine et al. [28] and Canales et al. [29] used the coupled CRTBP model to approximate the restricted four-body problem (RFBP) in the multi-moon planetary system. Fantino et al. [30] used a combination of multiple GAs and DSMs to reduce the relative arrival velocity for a mission to the four inner moons of Saturn. However, the investigation of PGA+AGA orbits in the RFBP is beyond the scope of this paper.

5. Conclusions

In this paper, an innovative flyby orbit combining the powered gravity assist (PGA) and aerogravity assist (AGA), called a PGA+AGA maneuver, was proposed and investigated in the planar elliptic restricted three-body problem (PERTBP). A PGA+AGA trajectory was divided into three portions: the PGA, AGA and ballistic portions, successively. In the PGA portion, continuous thrusts were conducted to decrease the altitude of a vehicle below the atmosphere edge. The optimal solver, GPOPS-II, was used to construct the PGA portion. In this paper, the AGA portion was composed of three phases, including the descent, level-flight and ascent phases. A simple flight-path angle guidance algorithm for the three phases was used to design the AGA portion. In the ballistic portion, we assumed that a vehicle was only affected by gravitational forces from celestial bodies. The PGA and AGA portions were the emphases of this paper.

Compared with a previous algorithm of the AGA orbit, the simpler three-stage guidance law had similar suboptimal results. Taking the Sun–Mars PERTBP system as an example, PGA+AGA orbits around Mars were constructed and discussed. Numerical results indicated that compared with original gravity assist (GA) orbits and even PGA orbits, PGA+AGA orbits were more various, and could obtain a larger velocity variation, energy variation or turn angle. Parameters of the AGA portion can significantly affect variations in energy, velocity and path. In addition, the influence of the true anomaly of the periapsis or the time of the periapsis was discussed to demonstrate the special characters of PGA+AGA orbits in the elliptic model. Numerical computation indicated that the epoch of periapsis of a vehicle had a noticeable influence on the velocity and energy variations for GA, PGA and PGA+AGA orbits, especially for PGA+AGA orbits. Therefore, it is necessary to investigate the PGA+AGA problem in the PERTBP.

In this paper, the AGA portion plays a dominant role in the change in velocity or energy, while the main role of the continuous thrust before periapsis in the optimal PGA process is set as a re-targeting of the altitude to take full advantage of the effect of AGA. Obviously, the global optimization including PGA and AGA can achieve more gain or loss of velocity or energy, but it is beyond the scope of this paper and will be our future work.