Optimal Gliding Trajectories for Descent on Mars

Abstract

In this paper, optimal gliding trajectories are analyzed, formulating an optimal control problem to be solved computationally via nonlinear programming. A multi-objective terminal cost function that minimizes the velocity, altitude, horizontal flight path angle, and the error of a desired terminal point is formulated. The results are validated via Monte Carlo simulations, analyzing the influence of the variations in the lift-to-drag ratios in the atmospheric entry. The results show feasible solutions for minimizing the distance to the desired final point, which have significant practical implications for the controlled gliding entry of a spaceplane on Mars. In this sense, the results also show that the longitudes of the landing points do not change too much for almost all trajectories, while the latitude of these points is in the interval of about 4 degrees for the majority of the trajectories. This suggests that it is possible to implement an optimal control approach with reasonable accuracy for the controlled gliding entry of a spaceplane on Mars, even in the presence of some uncertainties regarding the aerodynamic performance of the spacecraft, such as the lift-to-drag ratio.

1. Introduction

The terminal phase of an interplanetary mission to deploy a payload on the surface of Mars is composed of the entry, descent, and landing (EDL). This is a critical phase of the mission due to the spacecraft–atmosphere interaction, which increases aerodynamic and heat loads on the structure of the spacecraft due to deceleration from high hypersonic velocities, mainly dominated by drag. When compared to Earth’s atmosphere, the heat transfer and loads are lower on Mars due to its composition and the lower mass of the planet, which reduces the gravitational attraction at the descent. Spacecraft-like spaceplanes have been useful for re-entry on Earth, saving energy because they do not require the action of the propulsion system to glide, such as thrusters. At the same time, the spaceplanes are qualified for space activities; two examples of these are the space shuttles and the robotic X-37. Their large area-to-mass ratio (A/m), high aerodynamic performance, and the implementation of aerodynamic control surfaces allow gliding flight above the planet, increasing the range and cross-range of the trajectory. This property facilitates the direction of the spaceplane to a specific point for the final approach and landing, improving the controllability of the trajectory, different from the descent capsules, which do not have surfaces to increase the endurance of the atmospheric flight.

The optimization of the trajectory focuses on the satisfaction of a specific cost function along or at the end of the flight. A way to calculate optimal trajectories is through an optimal control problem (OCP). This method is highly implemented in the aerospace industry, robotics, and autonomous vehicles to compute the inputs of manipulated variables to control the state vector of the resulting trajectory to minimize or maximize the cost function. In the case of gliding spaceplanes, the idea is to find the history of the aerodynamic angles (angle of attack and banking angle) to direct the spacecraft onto the optimal path.

Various research has presented the benefits of optimal trajectories for interplanetary maneuvers passing over the atmosphere of a celestial body [1,2,3,4,5,6,7,8,9,10,11,12,13]. A way of optimizing the aeromaneuver of a hypersonic glider is via Pontryagin’s maximum, which is used to minimize the loss in energy, increasing the turn angle or atmospheric flight around Venus and Mars [4]. This is an example of optimization at a terminal time. Another kind of optimization, along the trajectory, was explored with the addition of a heat constraint, which reduces the velocity because it is proportional to the increase in heat [5,11]. When the trajectory is analyzed outside of the plane, control of the bank angle is required. This is advantageous in changing the inclination of the heliocentric orbit [6]. Other constraints were explored in [7], dealing with a traditional optimal control problem, different from [8], where multi-objective genetic algorithms were implemented, looking for a minimization of the Delta-V given by the propellant. To solve this problem, nonlinear programming (NLP) in an OCP was applied to optimal gravity-assist maneuver above Mars [1]. Optimal guidance was studied in [9]. The cost function to maximize the turn angle was presented in [10], and the optimization of a powered gravity assist was analyzed in [12]. Optimal trajectories above Venus, Earth, and Mars were studied to maximize the terminal velocity, latitude, and longitude, controlling the aerodynamic angles of attack and baking on a spaceplane at high altitudes at the beginning of the continuum flow [13,14].

Optimal gliding descent of hypersonic vehicles has been analyzed in some scientific papers; for instance, a second-order cone programming to optimize entry trajectories of hypersonic gliders with high lift/drag ratios was explored in [15]. Geographic constraints were applied in the entry guidance of autonomous gliders at high lift/drag, showing that the vehicle reached the target point by adapting to different kinds of missions [16]. Fast analytical solutions for a hypersonic glider with lift/drag greater than 3.0 were analyzed to update the calculated trajectory and to land in a specific cross-range [17]. A short-range guidance with a predictor–corrector algorithm was explored for fast re-entry [18]. Optimal lift control subject to initial and final values of the trajectory were derived from parametric studies in [19].

In addition to the large system (spacecraft) and according to the mission, sometimes the implementation of aerodynamic subsystems or components, such as aerodynamic decelerators, is required. Both inflatable aerodynamic decelerators and mechanically deployable systems are promising technologies enabling several benefits, including large payload delivery to the Martian surface or orbit and aerocapture missions to Venus or other planets. By expanding to a larger diameter prior to entry, such concepts allow higher entry performance than rigid aeroshells suitable for a variety of planetary or Earth high-speed entries requiring high-temperature materials [20]. Furthermore, being the ballistic coefficient at entry unconstrained by the launcher configuration, these deployable technologies provide operational flexibility over rigid aeroshells along with reduced aero-heating and surface pressure experienced during entry. A deployable Mars aeroshell concept was studied in [21] based on thermal protection system (TPS) panels fitted between retractable ribs. However, such systems typically suffer from uncontrolled re-entry and reduced trajectory accuracy due to the uncertainties during re-entry operations. Morphing technology has shown the potential to provide more accurate guidance trajectories and precise landing of space vehicles by using aerodynamic drag [22]. Other variations in optimal trajectories for this kind of vehicle, including ascent, powered descent, and additional phases of flight, are available in the scientific literature but are far from the focus of the present paper.

Different from the previous solutions explored in the literature, in this paper, the optimal control is selected (open loop) to find feasible trajectories from the spacecraft’s atmospheric entry point to a preselected final zone. However, the final approach or landing is not detailed in this research. The novelty of this paper is the sensitivity analysis of optimal trajectories during the atmospheric descent of a spaceplane on Mars, subject to uncertainties on lift-to-drag ratios (l/d), satisfying a multi-objective cost function at the same time. Following that goal, this paper is organized as follows: the dynamical model and the formulation of the cost problem are presented in Section 2. Section 3 shows the results of the simulations, including the discussion. Finally, the conclusions are listed in Section 4.

2. Formulation of the Optimal Control Problem

The formulation of the optimal control problem requires a mathematical model of the system, which in this case involves the spaceplane and Mars, the model of the control variables, the constraints, and the cost function. This section describes all of those parts.

2.1. Kinematics and Dynamics of the System

The state vector is represented by six nonlinear first-order differential equations, Equations (1)–(6). The origin of the inertial planetocentric reference system is on the center of an ideal spherical planet, with a homogeneous mass distribution and far from the influence of other bodies, such as moons and the Sun [1,14], also assuming a non-rotational planet. The deduction of the equations of motion is far from the purpose of this research; however, for a detailed mathematical demonstration, we invite the reader to follow the references [1,3,4,5,6,12,13,14,15,16].

The kinematic equations describe the rate of change in the radius ($R$), longitude ($\theta$), and latitude ($\phi$) as a function of the spaceplane velocity ($V$), flight path angle ($\gamma$), and azimuth. The three equations which describe the dynamics are a function of the aerodynamic forces of lift ($l$) and drag ($d$), the banking angle ($\beta$), the instantaneous mass of the spacecraft (m), and the acceleration of gravity as a function of the altitude ($g$).

$$\dot{R}=V\mathit{sin}\gamma ,$$

(1)

$$\dot{\theta }={\displaystyle \frac{V}{R}}\mathit{cos}\gamma \left({\displaystyle \frac{\mathit{sin}A}{\mathit{cos}\phi }}\right),$$

(2)

$$\dot{\phi }={\displaystyle \frac{V}{R}}\mathit{cos}\gamma \mathit{cos}A,$$

(3)

$$\dot{V}={\displaystyle \frac{d}{m}}-g\mathit{sin}\gamma ,$$

(4)

$$\dot{\gamma }=\left[\left({\displaystyle \frac{l}{m}}{\displaystyle \frac{\mathit{cos}\beta }{V\mathit{cos}\gamma }}-{\displaystyle \frac{g}{V}}\right)+{\displaystyle \frac{V}{R}}\right]\mathit{cos}\gamma ,$$

(5)

$$\dot{A}={\displaystyle \frac{l}{m}}{\displaystyle \frac{\mathit{sin}\beta }{V\mathit{cos}\gamma }}-{\displaystyle \frac{V}{R}}\mathit{cos}\gamma \mathit{tan}\phi \mathit{sin}A.$$

(6)

2.2. A Brief Description of the Aerodynamic

The aerodynamic accelerations due to lift (l/m) and drag (d/m) are a function of the spaceplane area-to-mass ratio (A/m), the relative velocity to the flow, its respective coefficients of lift (C_l) and drag (C_d), and the density ($\rho$) of the atmosphere, which is a function of the altitude. This last point is modeled as an exponential ($\rho ={\rho }_0{exp}^{(-h/H)}$), using the data available from the planetary data sheet [13,14,23], where the scale height ($H$) is 11.1 km and the density at the surface ${\rho }_0$ is 0.02 kg/m³. The aerodynamic accelerations are as follows:

$${\displaystyle \frac{l}{m}}=0.5C_l{\displaystyle \frac{A}{m}}\rho V^2,$$

(7)

$${\displaystyle \frac{d}{m}}=0.5C_d{\displaystyle \frac{A}{m}}\rho V^2.$$

(8)

The lift and drag coefficients were modeled from the data of the X-34 spaceplane in continuum flow [24], which, in this case, is selected only to be a function of the angle of attack (AOA). The effective angles for the control are from −4 deg to 28 deg, and the maximum performance is reached at 11 deg, generating a l/d of 2.2, as presented in Figure 1.

2.3. Path Constraints

It is assumed that the spaceplane begins its atmospheric descent at 92.0 km of altitude, with a negative value of the flight path angle, meaning that one component of its velocity vector is pointing towards the center of the planet. This part of the descent begins after orbital decay, which could be calculated to be in a specific region, also with aerodynamic control; an example is presented in [25]. Due to the nature of the descent, where the spacecraft loses its mechanical energy due to drag, it is expected to have a decrease in velocity. For those reasons, the constraints on the state vector, the control vector, and the path, in an interval of time between (t₀) and (t_f), are selected to be as follows:

$$\left.\begin{array}{c}{t}_0<t\le t_f\\ 50s\le t_f\le 800s\\ \begin{array}{c}{R}_{Planet}\le R\left(t\right)<R_{Planet}+92km\\ 0.0m/s\le V\left(t\right)\\ \begin{array}{c}-180deg\le \theta \left(t\right)\le 180deg\\ -90deg\le \phi \left(t\right)\le 90deg\\ \begin{array}{c}-20deg\le \gamma \left(t\right)\le 20deg\\ 0deg\le A\left(t\right)\le 359.9deg\\ \begin{array}{c}-4deg\le \alpha \left(t\right)\le 28deg\\ -180deg\le \beta \left(t\right)\le 180deg\\ \dot{\alpha }\left(t\right),\ddot{\alpha }\left(t\right),\dot{\beta }\left(t\right),\ddot{\beta }\left(t\right)=\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}$$

(9)

2.4. Cost Functions

The descent problem focuses on reducing the velocity and altitude for a specific terminal point at the final time (t_f). An aleatory point of 26 deg in longitude and 35 deg in latitude is selected as the terminal point. For those reasons, the specified cost function is composed of four terminal costs in order to minimize its value at the final time. So, the OCP is formulated to minimize the following:

$$\underset{\mathbf{U}\left(t\right),t_f}{\min }\mathbf{J}\left(t_f\right);\mathbf{J}\left(t_f\right)={\mathbf{J}}_1\left(t_f\right)+{\mathbf{J}}_2\left(t_f\right)+{\mathbf{J}}_3\left(t_f\right)+{\mathbf{J}}_4\left(t_f\right),$$

(10)

for the objective functionals

$$\left.\begin{array}{c}{\mathbf{J}}_1\left(t_f\right)=V\left(t_f\right)\\ {\mathbf{J}}_2\left(t_f\right)=R\left(t_f\right)\\ \begin{array}{c}{\mathbf{J}}_3\left(t_f\right)=‖\gamma \left(t_f\right)‖\\ {\mathbf{J}}_4\left(t_f\right)={\left(\theta \left(t_f\right)-26\right)}^2+{\left(\phi \left(t_f\right)-35\right)}^2\end{array}\end{array}\right\}$$

(11)

subject to the dynamics of the system, Equations (1)–(8), and the path constraints (9). The initial conditions are presented in Table 1.

3. Results

The OCP formulated in Section 2 is solved via NLP using the optimization suit GEKKO in Python, which includes the interior point optimizer (IPOPT) as a solver [26]. This software has been used successfully to solve OCP and dynamic optimization problems, finding optimal trajectories; for instance, it was implemented to optimize high-velocity gravity-assist maneuvers for spaceplanes [14]. The maximum number of allowed iterations is selected to be 300, the objective function tolerance lower than 1.0 × 10⁻⁹, and the relative tolerance 1.0 × 10⁻¹⁰. The mean computational time to solve the problem is less than 30 s on a PC with a Core i5 (12th) and 16 GB of RAM. The library Matplotlib on Python was used to generate the graphics presented in this paper [27].

The initial conditions of the state vector, after the orbital decay, in a descent at the beginning of the continuum flow are assumed to be the following:

Table 1. Initial conditions.

Initial time	0.0 s
Altitude	92.0 km
Latitude	0.0 deg
Longitude	0.0 deg
Velocity	6.5 km/s
Flight path angle (FPA)	−4.0 deg
Azimuth	90.0 deg
Mass	8200.0 kg
Area	332.0 m2

Table 1. Initial conditions.

Initial time	0.0 s
Altitude	92.0 km
Latitude	0.0 deg
Longitude	0.0 deg
Velocity	6.5 km/s
Flight path angle (FPA)	−4.0 deg
Azimuth	90.0 deg
Mass	8200.0 kg
Area	332.0 m2

3.1. Nominal Trajectory

The first feasible solution, or optimal solution, was found for a final time of 717.49 s, terminal velocity of 72.65 m/s, a final altitude of 0.0 m, 25.71 deg in longitude, and 28.96 deg in latitude. The behavior of the state variables and the control variables along the trajectory is presented in Figure 2, Figure 3, Figure 4 and Figure 5.

Figure 5 provides a detailed view of the nominal trajectory. Looking first at the altitude behavior, we see that the spacecraft initially falls very fast. Then, it reaches an altitude where the density of the atmosphere is higher, and the spacecraft bounces back, increasing its altitude. After that, it starts to fall again until arriving at the surface of Mars.

The velocity has two fast-decreasing zones, at the beginning and the end of the trajectory, during the fall phases. They are separated by a long phase of slight decrease in velocity, which occurs when the spacecraft bounces back due to the crash in a dense region of the atmosphere. Azimuth, flight path angle, and bank angle all have swift motions at the end of the trajectory, where the atmosphere is denser and the spacecraft is working to reach zero velocity when approaching the surface of Mars, at the same time that it is reducing the distance to the final point. Latitude and longitude have smooth evolutions from the initial point to the specified point of landing. In contrast, the angle of attack has fast motion at the beginning and end of the trajectory, with a large phase of near-constant value in the middle.

3.2. Uncertainties on Aerodynamic Coefficients: Sensitivity Analysis

The planetary entry and descent are complex aeromaneuvers, because they require high precision, which is difficult due to the estimated uncertainty of some factors, such as the behavior of the atmospheric conditions. The atmosphere is a stochastic system that depends on solar activity, the magnetic field of the planet, and other variables. On the other hand, factors like uncertainties in the state vector and/or in the aerodynamic performance could reduce the precision of the controller. In order to analyze that, the changes in l/d on the trajectory were analyzed from a mean value of 0.96, assuming a normal distribution with 0.01 on standard deviation.

In this study, 100 random values for the spacecraft’s l/d ratio were generated, and the simulations were repeated for each value. The main goal is to see how this parameter affects the spacecraft’s trajectories, particularly the location of the final or terminal point. Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 show the results.

Looking at all the figures, Figure 6 confirms that all the trajectories ended with the spacecraft on the surface of Mars. Thus, no escapes from the planet were obtained in the simulations, which means that the variations in the l/d ratio were not strong enough to cause the spacecraft to either leave Mars or collide.

Figure 7 shows in detail the final azimuth and flight path angles. We see that most of the trajectories ended with flight path angles near −18.6 degrees, so the variations in the l/d ratio did not affect this variable much. The opposite occurs for the final azimuth angle. They have a large dispersion, going from zero to more than 100 degrees. So, this variable is very sensitive to the l/d ratio, and it is impossible to make accurate predictions of this value without knowing the l/d ratio of the spacecraft very well.

Figure 8 is the most important one for the goals of the present research. It shows the location of the final points. The final latitude is around 30 degrees for almost all trajectories. The final longitude goes closer to 26 degrees for the majority of the trajectories, which is a better interval compared to the results obtained for the latitude. So, in general, the variations in the l/d ratio change the final point but keep them in a square of 5 by 2 degrees in terms of latitude and longitude. In a more detailed maneuver, this final approach is selected to begin at a higher altitude so that there is enough time to loiter and make the final adjustment before the landing. After that, systems like retrorockets, parachutes, or skids (used on the X-15) could be deployed.

Figure 9 indicates that almost all the trajectories (in different colors) have similar behaviors when compared to the nominal one in terms of velocity and altitude. With very few exceptions, they start the motion towards Mars, bounce back when reaching a point of higher density in the atmosphere, and then the spacecraft falls again. The velocity has a slow decreasing phase during this bouncing motion and a fast decrease when out of this zone.

Figure 10 shows the different trajectories that each value of l/d generates for the spacecraft. We see a large concentration of solutions near the nominal trajectories, not only at the landing point but also throughout the motion.

The angle of attack (Figure 11) is also variable. Almost all the trajectories follow the nominal trajectory, and variations in l/d have lower effects. The banking angle (Figure 12) has a more dispersed evolution, with trajectories that differ considerably with a slight variation in l/d, adjusting the direction of the flight closer to the final point.

In summary, the OCP results were presented in this Section, obtaining feasible trajectories that satisfied the objectives and constraints (Equations (9)–(11)). At the same time, a sensitivity analysis from Monte Carlo simulations was performed, calculating optimal solutions from changes in the values of the lift and drag coefficients. The optimal control problem was formulated and solved via nonlinear programming to calculate the control history of optimal gliding descent above Mars. The results show feasible solutions, minimizing the four cost functions at a terminal time. The results showed that all the trajectories landed on Mars, so neither escapes from the planet nor collisions on its surface occurred for the l/d values used.

We also saw that most of the trajectories ended with flight path angles near −18.6 degrees, but the final azimuth angle had a large dispersion, from zero to more than 100 degrees. So, the final azimuth angle is very sensitive to the l/d ratio, and it is impossible to make an accurate prediction of this value without knowing the spacecraft’s l/d ratio.

4. Conclusions

The research performed here showed the effects of the variation in the l/d ratio in the descent trajectories of a spacecraft on Mars. It can be used to guide the preliminary design of missions that aim to use a gliding fly in the atmosphere of Mars to descend a spacecraft close to the surface of this planet. The most important result of this research is that the feasible trajectories landed at latitudes around 30 degrees and longitudes close to 26 degrees for the majority of the trajectories. Even though the latitude is lower than expected, it is still not wholly unpredictable. Regarding the trajectories of the spaceplane, it was shown that most of them behaved similarly to the nominal trajectory, both in terms of velocity and altitude time evolution. Almost all those trajectories bounced back when reaching a point of higher density in the atmosphere of Mars, and after that, the spacecraft fell again. At the same time, the velocity had a slow decreasing phase during this bouncing motion and a fast decrease when out of this zone. The trajectories obtained for different values of l/d were very similar to the nominal trajectories during the whole motion. The angle of attack has a similar behavior, while the banking angle had a more dispersed evolution, compensating for the losses in lift due to lower density to reach the desired point.