Integrated Power and Propulsion System Optimization for a Planetary-Hopping Robot

Abstract

Missions targeting the extreme and rugged environments on the moon and Mars have rich potential for a high science return, although several risks exist in performing these exploration missions. The current generation of robots is unable to access these high-priority targets. We propose using teams of small hopping and rolling robots called SphereX that are several kilograms in mass and can be carried by a large rover or lander and tactically deployed for exploring these extreme environments. Considering that the importance of minimizing the mass and volume of these robot platforms translates into significant mission-cost savings, we focus on the optimization of an integrated power and propulsion system for SphereX. Hydrogen is used as fuel for its high energy, and it is stored in the form of lithium hydride and oxygen in the form of lithium perchlorate. The system design undergoes optimization using Genetic Algorithms integrated with gradient-based search techniques to find optimal solutions for a mission. Our power and propulsion system, as we show in this paper, is enabling, because the robots can travel long distances to perform science exploration by accessing targets not possible with conventional systems. Our work includes finding the optimal mass and volume of SphereX, such that it can meet end-to-end mission requirements.

1. Introduction

Exploring the far corners of the solar system by performing orbital, surface, and subsurface exploration with the help of human and robotic explorers will define the next wave of space exploration. These exploration platforms will reach diverse surface environments on the moon and Mars, including caves, canyons, cliffs, skylights, and craters. They will answer fundamental questions about the origins of the solar system, origins of life, and prospects for In situ Resource Utilization (ISRU). These are some of the high-priority targets, as outlined in [1]. Some of these environments, such as caves and lava tubes, are well insulated from radiation, harsh weathering such as dust storms, and high and low external temperatures. These insulated environments could harbor isolated, ancient ecosystems or remains of the last survivors. They can also provide insight as a potential habitat for future human explorers. Current robotic missions are unable to reach these places of interest, mainly due to precision landing limitations, an inability to reach or move in rugged environments, and the risks involved in adverse mission culture. This is despite the great potential science that is possible from taking these exploratory risks. Hence, there is an important need for novel robotic systems that can access and explore these rugged environments [2]. A credible solution is to develop mother–daughter architectures that permit daughter crafts to take high exploratory mission risks for high reward science but without putting the mothercraft or the rest of the mission in jeopardy. One such example is the Mars helicopter, Ingenuity, that hitched a ride to Mars on the Perseverance rover and demonstrated technologies to test powered, controlled flight on another world for the first time [3].

Here we present a small, modular, low-cost, spherical robot platform called SphereX that is designed to explore rugged environments such as caves, lava tubes, pits, and canyons on the moon, Mars, icy moons, comets, and asteroids. SphereX operates by hopping and rolling short distances [4,5]. SphereX is designed with space-grade electronics, including a command and data-handling board, power management/distribution board, and radio with S/X-band antennas for communication/coordination among teams of robots and with a mothercraft. Mobility is enabled using a single thruster-propulsion system combined with a 3-axis reaction wheel-based attitude control to perform ballistic hops. Rolling is enabled using the attitude control-system alone. The onboard power supply contains a power generation, storage, and distribution subsystems. The remaining volume contains a stereo camera and a LiDAR system for 3D imaging, surveying, and navigation. A flagship rover or lander can carry several SphereX robots that can be deployed on demand to access and explore extreme environments that would otherwise be inaccessible to the rover or lander. Here, we propose an integrated power supply and propulsion system for SphereX consisting of Proton-exchange membrane (PEM) fuel cells for power and a H₂/O₂ propulsion system as shown in Figure 1. Hydrogen is generated on demand through the reaction of water and lithium hydride (LiH) [6]. Oxygen is generated on demand through a catalytic decomposition of lithium perchlorate (LiClO₄) [7]. The hydrogen fuel and oxidizer are transported to the fuel cell system using micropumps while pressurized nitrogen gas is used to transport the reactant to the propulsion combustion chamber. Wastewater from the PEM fuel cell is collected and recycled to generate hydrogen on demand using the lithium hydride (LiH) hydrogen generator. Based on lessons learned from our earlier research [8], we intend to implement a fuel cell/battery hybrid system to maximize mission life. Here, our work focuses on finding the optimal mass and volume of SphereX that meet the constant power demand for a predefined mission life while exploring a target distance. There is an important need to optimize the mass and volume of SphereX due to the high launch cost involved in integrating planetary exploration robots, instruments, and platforms.

Finding the optimal design of SphereX in terms of volume and mass (which in turn impacts cost and mission life) is a highly coupled problem between multiple disciplines.

Traditionally, space systems are optimized through the rigorous evaluation of each discipline independently. Although feasibility is achieved with this labor-intensive approach, there is no guarantee for achieving an optimal overall system. Thus, the design of SphereX could benefit from the application of Multidisciplinary Design Optimization (MDO). However, complexity arises in an MDO approach to design SphereX due to challenges in modeling, the use of structurally complex and discontinuous objective cost function and design space, and a wide range of constraints. Here, we approach this problem by applying a hybrid optimization method by applying Genetic Algorithm (GA)-based optimization integrated with multiple gradient-based optimization sequences [9]. The methodology used in this research can find near optimal designs in terms of mass, volume, and assembly feasibility for SphereX applied to different missions. Before going into the next sections, we first present our main contributions in this paper.

• We propose an integrated power supply and propulsion system for SphereX consisting of Proton-exchange membrane (PEM) fuel cells for power and a H₂/O₂ propulsion system.
• We develop a system-level optimization problem to find optimal design solutions of SphereX in terms of mass, volume, and power.
• A comparative analysis of the optimal design solution of SphereX is provided against other relevant systems.

In the following sections, we present background information on fuel cells, miniaturized propulsion systems, and multidisciplinary optimization approaches used in the past (Section 2). This is followed by Section 3, where we present the details of all the subsystems of SphereX, along with the formulation of the system-level optimization problem and details of the optimization algorithm used. Section 4 presents the results found using the proposed algorithm for two different mission scenarios, which is followed by a comparative analysis between the proposed system and other similar systems in Section 5. Section 6 presents the conclusion of the paper.

2. Background

Fuel cells have been suggested for a multitude of autonomous robotic systems and sensor network modules such as for powering hopping robots [10], ground robots [11], humanoid robots [12], unmanned underwater vehicles (UUVs) [13], and landers and rovers for moon and Mars exploration [14,15]. Fuel cells being clean, quiet, and offering high efficiency and specific energy makes them an ideal option as a power source for robotic systems to be deployed in pristine environments such as caves and lava tubes on the surface of the moon and Mars. However, factors such as the degradation of fuel-cell life due to fluctuations in power demand and the efficient storage of hydrogen and oxygen prevents it from becoming a widespread energy source [8]. One proposed application demonstrates a hybrid system where a battery is recharged with the fuel cell, which is used to power the system [8,16]. For robot systems such as SphereX, which is small in mass and volume, the cryogenic storage of hydrogen and oxygen at high pressures is not practical. For such applications, storing hydrogen and oxygen in solid form is viable and has been considered before. One method is to release hydrogen from solid hydride through depressurization; however, it remains unattractive, due to low storage efficiencies in the order of 0.5% to 2.5% [17]. Hydrides that use heat to release hydrogen can reach up to 18% storage efficiency, but substantial energy and infrastructure are required to reach temperatures of 70 to 800 °C [17]. Another approach is to produce hydrogen through the hydrolysis reaction of metal hydrides with water. Similarly, oxygen can be stored in the form of metal oxides. The Soyuz spacecraft used chemical oxygen generators with potassium superoxide, while the International Space Station (ISS) and Mir used a Vika oxygen generator with lithium perchlorate [18].

Miniaturized propulsion technology for space systems has relied heavily on low-specific-impulse and low-thrust technologies. However, the propulsion system for SphereX would require producing a thrust greater than its weight in order to perform ballistic hopping on the surface of the moon and Mars. Cold gas or pulsed plasma systems are fairly well established and demonstrated for small delta-v missions, but the specific impulse and thrust developed are fairly low. A liquid sulfur hexafluoride cold gas thruster has been used in a 3U CubeSat mission Can X-2 and attained a specific impulse of 50 s [19]. The propulsion system for the MarCo Mission uses a R-236fa refrigerant with a specific impulse of 40 s [20]. Electrospray propulsion systems have not only achieved significant maturity, but also have the disadvantage of low thrust in the order of 1mN [21]. Solid rocket motors are a good option for SphereX, due to their overall simplicity, technology maturity, and high specific impulse and thrust. However, for performing multiple hops with SphereX, solid rockets are not ideal, as they expend all their propellant at once in addition to producing vibrations and have difficulty in throttling. A compelling alternative is to use Polymer Electrolyte Membranes to electrolyze water into hydrogen and oxygen on-demand and to produce high thrust and specific impulse through combustion. An electrolysis propulsion system was designed that offers a specific impulse of 400 s, but the power required is between 50 to 200 W [22]. Another electrolysis propulsion system called Hydros was developed that can deliver up to 0.25–0.6 N of thrust at 258 s of specific impulse [20]. However, major challenges in electrolysis propulsion systems remain in preventing the water from freezing and a high-power requirement.

With an overview of fuel cells and a miniaturized propulsion system described above, designing them together for a robotic system to meet mission requirements is a complex task. Aerospace system designers have been using multidisciplinary design optimization (MDO) approaches to a variety of aerospace problems for decades now. However, typically, most of the efforts have been focused on designing aircraft structures and space launch vehicles. Its application to miniaturized space systems such as planetary exploration robots is minimal. The launch vehicle design completed by Olds et al. [23] is the first identified MDO application in space systems. MDO was first applied to design satellites by Matossian as demonstrated when designing an Earth-observing satellite mission [24]. Later, the application of MDO to develop space-system engineering tools was continued by Riddle [25], Bearden [26], George et al. [27], Fukunaga et al. [28], Mosher et al. [29], Stump et al. [30], and Barnhart et al. [31]. Ravanbakhsh et al. [32] worked on structural design-sizing tool using an MDO approach. An evaluation of optimization techniques was provided by Taylor et al. [33] applied to complex spacecraft design problems. A multi-objective, multidisciplinary design optimization method for space systems was developed by Jilla et al. [34]. The application of the distributed Collaborative Optimization (CO) method was shown by Jafarsalehi et al. [35] for small-satellite missions. The method used gradient-based techniques at the discipline level and gradient-free GA at the system level. The MDO literature provided multiple instances where system engineering tools were developed using mathematical optimization techniques to design complex systems comprising multiple disciplines.

Motivated by these ideas, our approach includes designing a combined power and propulsion system for SphereX by storing hydrogen and oxygen in solid form. Moreover, we use an MDO approach to optimize the overall design of SphereX considering predefined mission requirements.

3. System Design

This section provides a detailed description of each subsystem of SphereX along with the design specification used. First, we provide a detailed description of the fuel cell subsystem, followed by the propulsion subsystem, and then the hydrogen and oxygen generators, followed by the storage tanks and other relevant subsystems. With all the subsystems described, we then present the formulation of the system-level optimization problem along with details of the genetic algorithm used to find optimal solutions for our problem.

3.1. PEM Fuel Cells

The working principle of PEM fuel cells used as a power source is discussed in detail in Appendix A. The goal of this optimization effort is to find the optimal operating voltage $V$ and current density $i$ of the fuel cell system, such that the number of cells $𝓃$ and the mass of oxygen $m_{O_2,fc}$ and hydrogen $m_{H_2,fc}$ used is minimized, satisfying a constant power demand $P$ for a mission lifetime $\Gamma$. The design variables are ${\mathsf{𝕩}}_{fc}=\left[𝓃,i\right]$. The values of $𝓃$, $m_{O_2,fc}$, and $m_{H_2,fc}$ are normalized between [0, 1], represented by $\underset{\_}{𝓃}$, ${\underset{\_}{m}}_{O_2,fc}$ and ${\underset{\_}{m}}_{H_2,fc}$ based on their upper and lower bounds, and the objective is a weighted sum of the normalized values. The power generated $P_{fc}$ by the fuel cell system is the product of the total voltage of the fuel cell stack $V_{fc}=𝓃V$ and the current $I_{fc}=𝓃ai$, where $a$ is the MEA surface area of each cell. One constraint is added to the optimization problem, such that the power generated $P_{fc}$ is greater than or equal to the power demand $P$. The optimization problem is mathematically described as:

$$\begin{array}{c}min{f}_{fc}\left({\mathsf{𝕩}}_{fc}\right)={\alpha }_1\underset{\_}{𝓃}+{\alpha }_2{\underset{\_}{m}}_{O_2,fc}+{\alpha }_3{\underset{\_}{m}}_{H_2,fc}\end{array}$$

(1a)

$$\begin{array}{c}\begin{array}{c}subject to \{g_{fc}\left({\mathsf{𝕩}}_{fc}\right)\equiv P_{fc}\ge P\end{array}\end{array}$$

(1b)

The optimization problem defined here is a mixed-integer optimization problem, as the design variable $𝓃$ can take only integer values. Thus, we use genetic algorithms to solve the optimization problem. Figure 2 shows the value of the objective function over the number of generations with the weights ${\alpha }_1={\alpha }_2={\alpha }_3=1$, MEA surface area $a=10{ \mathrm{cm}}^2$, and power demand $P=16 \mathrm{W}$. The optimal values of the design variables are the number of cells $𝓃=3$, the current density $i=194.38 \mathrm{mA}/{\mathrm{cm}}^2$, and the operating voltage $V=0.9146 \mathrm{V}$.

3.2. Propulsion System

The dynamics of the robot while using the combined action of the propulsion system and the attitude-control system for performing ballistic hops is presented in Appendix B. To perform that, the critical subsystem for SphereX is the propulsion system. Each SphereX contains one primary lift engine at the bottom that uses H₂ fuel and O₂ oxidizer. Given that SphereX is limited in mass and volume, the propulsion system uses nitrogen gas to transport the reactants into the combustion chamber, hence avoiding the use of pumps and mechanical devices. The thermodynamic parameters of the combustion products, namely combustion temperature $T_c$, molar mass $M$, and ratio of specific heats $\gamma$, were determined using the Cpropep software for a given combustion pressure $p_c$ and mixture ratio $r_m=m_{O_2, prop}/m_{H_2, prop}$ [36]. Figure 3 shows the variation of combustion temperature, molar mass, and ratio of specific heats against the combustion pressure, mixture ratio, and the corresponding specific impulse for an exit pressure of 200 Pa. An experiment was performed to collect data by using the Cpropep software, and a model was fitted using a radial basis function. The thermodynamic model is then used for optimizing the propulsion subsystem.

For the robot (SphereX) with mass $𝓂$ and radius $𝓇$ to perform ballistic hopping on the surface of a planet with gravity $g$, a propulsion system is required to generate thrust, such that $\left|\right|F\left|\right|>𝓂g$. For our analysis, we have considered $\left|\right|F\left|\right|=2𝓂g$, and we need to design each element of the propulsion system. Figure 4 shows the basic elements of cylindrical combustion chamber and conical nozzle. For the propulsion system to produce a constant thrust of $\left|\right|F\left|\right|$, the throat area of the nozzle is:

$$\begin{array}{c}{A}_{th}=\frac{\left|\right|F\left|\right|}{p_c\sqrt{\frac{2{\gamma }^2}{\gamma -1}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}+\left(p_e-p_a\right)ϵ}\end{array}$$

(2)

The theoretical nozzle expansion ratio $ϵ$ is then expressed as:

$$\begin{array}{c}ϵ=\frac{A_e}{A_{th}}=\frac{\mho }{\sqrt{\frac{2\gamma }{\gamma -1}{\left(\frac{p_e}{p_c}\right)}^{\frac{2}{\gamma }}\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\left(\gamma -1\right)}{\gamma }}\right]}}\end{array}$$

(3)

where, $\mho =\sqrt{\gamma {\left(2/\left(\gamma +1\right)\right)}^{\left(\gamma +1\right)/\left(\gamma -1\right)}}$ is the Vandenkerckhove function [37]. For a conical nozzle, the nozzle throat section is defined by its radius, convergent cone half, and divergent cone half angle. The radius $r_n$ ranges from 0.5 to 1.5 times the throat radius, the convergent cone half angle $\beta$ ranges from 20° to 45°, and the divergent cone half angle $\alpha$ ranges from 12° to 18°. The length of the diverging and converging conical nozzle section can then be expressed as:

$$L_{n\left(div\right)}=\frac{r_{th}\left(\sqrt{ϵ}-1\right)+r_n\left(\sec \alpha -1\right)}{\tan \alpha }; L_{n\left(con\right)}=\frac{r_{th}\left(\sqrt{{ϵ}_c}-1\right)+r_n\left(\sec \beta -1\right)}{\tan \beta },$$

(4)

where, $r_{th}$ is the radius of the nozzle throat, and ${ϵ}_c=A_c/A_{th}$ is the contraction area ratio (where, $A_c$ is the area of the combustion chamber). For a propellant with characteristic length $L^{*}$, the time needed for vaporization and the complete reaction of the propellants, known as residence time, is ${\tau }^{*}=L^{*}{c}^{*}/R{T}_c$. The minimum chamber volume ${\mathcal{V}}_c$ required can then be determined as, ${\mathcal{V}}_c={\tau }^{*}\dot{m}R{T}_c/p_c$, where, $\dot{m}=A_{th}{p}_c\sqrt{\gamma {\left(2/\left(\gamma +1\right)\right)}^{\left(\gamma +1\right)/\left(\gamma -1\right)}/R{T}_c}$ is the mass flow rate. Finally, the length of the combustion chamber can be calculated as $L_c={\mathcal{V}}_c/A_c$. Moreover, the length of the injector $L_{inj}$ is estimated to be 25% of the sum of the length of the nozzle and the combustion chamber. As such the total length is defined as $L_{total}=L_{n\left(div\right)}+L_{n\left(con\right)}+L_c+L_{inj}$. Considering the combustion chamber to be a cylindrical shell, the wall thickness is, $t_c=FS{p}_c{r}_c/\sigma$, where $r_c$ is the radius of the combustion chamber, $\sigma$ is the tensile strength of the material used, and $FS$ is a safety factor. The thickness of the converging and diverging section of the nozzle is assumed to be the same as the combustion chamber wall thickness. The mass of the combustion chamber and nozzle $m_{c+n}$ is then calculated by multiplying the total surface area $S_{c+n}$ with the wall thickness $t_c$ and density of material used $\rho$ as $m_{c+n}=S_{c+n}{t}_c\rho$. Moreover, the mass of injectors, sensors, control equipments, valves, fittings, pipes, and other necessary interfaces are estimated as 20% of the propulsion system’s dry mass $m_{dry}$.

Next, we calculate the mass of propellant required for the robot to explore a target distance $d_{targ}$. The delta-v, $\mathsf{\Delta }{v}_{hop}$, required for the robot to perform a single hop of distance $d_{hop}$ is calculated by solving the optimal control problem for ballistic hopping as discussed in Section 4. Hence, the robot needs to perform $n=d_{targ}/d_{hop}$ hops, and, as such, the total delta-v required to explore the target distance is $\mathsf{\Delta }{v}_{total}=n\mathsf{\Delta }{v}_{hop}$. The mass of propellant required is then calculated as $m_{prop}=𝓂\left(1-e^{-\mathsf{\Delta }{v}_{total}/I_{sp}{g}_0}\right)$, where the specific impulse $I_{sp}$ is

$$I_{sp}=\frac{1}{g_0}\left(\sqrt{2R{T}_c\left(\frac{\gamma }{\gamma -1}\right)\left[1-{\left(\frac{p_e}{p_c}\right)}^{\frac{\gamma -1}{\gamma }}\right]}+\frac{\left(p_e-p_a\right)A_e}{\dot{m}}\right).$$

(5)

Thus, the mass of the oxidizer and fuel is calculated as $m_{O_2,prop}=r_m{m}_{prop}/\left(r_m+1\right)$ and $m_{H_2,prop}=m_{prop}/\left(r_m+1\right)$.

Optimization

The objective of the optimization process for the propulsion subsystem is to minimize the total mass of the propulsion system, $m_{p\left(total\right)}=m_{dry}+m_{prop}$. The design variables are combustion pressure $p_c$, exit pressure $p_e$, mixture ratio $r_m$, and contraction area ratio ${ϵ}_c$, ${\mathsf{𝕩}}_p=\left[p_c,p_e,r_m,{ϵ}_c\right]$. One constraint is added to the optimization function such that the total length $L_{total}$ is less than 90% of the radius of the robot. The mass $m$ and radius $r$ of the robot, target exploration distance $d_{targ}$ and single hopping distance $d_{hop}$ should be given as the input. The optimization problem is the following:

$$\begin{array}{c}min{f}_p\left({\mathsf{𝕩}}_p\right)=m_{p\left(total\right)}\end{array}$$

(6a)

$$\begin{array}{c}subject to: g_p\left({\mathsf{𝕩}}_p\right)\equiv L_{total}<0.9𝓇\end{array}$$

(6b)

The design problem is modeled as a nonlinear optimization problem (NLP), and a sequential quadratic programming (SQP) method is used to solve it. At each iteration of SQP, the gradients and the hessian of the objective function and the constraints need to be calculated. It is performed using the finite difference method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method respectively. The variations of the objective function value, constraint violation, and the first-order optimality of the optimization are shown in Figure 5. The objective function value remains stationary, and the constraint violation and the first-order optimality approach zero.

3.3. Hydrogen Generator

One of the major factors preventing H₂ from becoming a widespread energy source is the challenges involved in efficient storage. The cryogenic storage of H₂ at high pressure is not practical for small, low-power robotic systems such as SphereX because of complexity and difficulty in integrating them in such a small volume. Metal hydrides are considered for SphereX because of easy storage capabilities and their high gravimetric hydrogen weight densities. Metal hydrides react with water through the process of hydrolysis to release hydrogen and produce a metal hydroxide, some of which are shown in

Table 1. Gravimetric hydrogen content of various metal hydrides.

Hydride	Hydrogen Weight Content
LiBH4	18.4%
NaBH4	10.6%
Be(BH2)4	20.8%
NaAlH4	7.4%
NaH	4.1%
CaH2	4.7%
LiH	12.5%

.

Among the metal hydrides considered, the hydrogen production rates for sodium borohydride and lithium borohydride are slow and require expensive catalysts for reaction completion. The hydrogen content of beryllium borohydride is very high, but it is too reactive and toxic. Alkali metal hydrides such as lithium hydride, sodium hydride, and calcium hydride can produce hydrogen on contact with water and do not require a catalyst. LiH has the highest hydrogen weight content and is selected for SphereX. LiH is composed of lithium metal bonded with hydrogen and reacts with liquid water or water vapor to produce H₂ gas and LiOH according to the following reaction [38,39]:

LiH + H₂O → LiOH + H₂

Experiments have shown that the above reaction can achieve 95–100% reaction-completion rates with an excess of water [38,39]. The molar mass of hydrogen is $2.014\times {10}^{-3}$ kg/mole, and that of lithium hydride is $7.95\times {10}^{-3}$ kg/mole. Thus, the amount of lithium hydride required is:

$$\begin{array}{c}{m}_{LiH}=3.9435\left(m_{H_2,fc}+m_{H_2,prop}\right)\end{array}$$

(7)

Similarly, for each mole of H₂ produced, one mole of H₂O is required. The PEM fuel cell will produce water at a rate of $\dot{H_{2}O}=9.34\times {10}^{-8}P/V \mathrm{kg}/\mathrm{s}$, which will be reused for generating H₂ from LiH. The molar mass of water is $18.015\times {10}^{-3}$ kg/mole, thus the amount of water required is:

$$\begin{array}{c}{m}_{H_{2}O}=8.9373\left(m_{H_2,fc}+m_{H_2,prop}\right)-\dot{H_{2}O}\Gamma ,\end{array}$$

(8)

where, $\Gamma$ is the mission lifetime.

3.4. Oxygen Generator

In the absence of atmospheric oxygen for operating the fuel cells, oxygen must be stored for both the fuel cells and propulsion system. Similar to hydrogen storage, the high pressure and cryogenic storage of oxygen is also not practical for small, low-power applications such as SphereX. However, chemical oxygen generators can be used that store metal oxides. When combined with a catalyst under sufficient temperature, the resulting chemical reaction releases oxygen and produces heat.

Table 2. Gravimetric oxygen content of various metal oxides.

Oxide	Oxygen Weight Content
NaClO3	45.1%
BaO2	18.9%
KClO4	46.2%
KO2	45.0%
LiClO4	60.1%
BeCl2O8	61.5%

shows some examples of metal oxides as a source of oxygen.

Oxygen candles produce oxygen by igniting a mixture of sodium chlorate and iron powder at about 600 °C. Oxygen generators used in commercial airlines contain a mixture of sodium chlorate, 5% barium peroxide, and 1% potassium perchlorate, which releases oxygen once ignited at 260 °C. Soyuz spacecraft used chemical oxygen generators with potassium superoxide, where KO₂ reacts with H₂O and CO₂ to produce oxygen. Another proposed solution to produce oxygen is to use tetramethylammonium ozonide because of its low molecular weight and 39% oxygen content. In terms of oxygen weight content, lithium perchlorate has one of the highest and has already been used in the Vika oxygen generator on Mir and the International Space Station (ISS). Although beryllium perchlorate has the highest oxygen weight content, it is expensive and toxic. The thermal decomposition of lithium perchlorate at 400 °C results in a 60% by-weight release of oxygen according to the following reactions:

Main reaction: LiClO₄ → LiCl + 2O₂; side reaction: 4LiClO₄ → 2Li₂O + 7O₂ +2Cl₂

The fast decomposition of pure LiClO₄ as seen in the main reaction requires a temperature of 400 °C. However, the high temperature leads to the weak development of the side reaction. Hence, some catalysts are added in order to increase the LiClO₄ decomposition rate. Transition-metal oxides such as Fe₂O₃, Co₂O₃, TiO₂, MnO₂, and CuO increase the decomposition rate of perchlorates. Moreover, Li₂O₂ and Li₂O have been studied for the catalyzed decomposition of LiClO₄ [40]. From the main reaction of decomposition of LiClO₄ for two moles of oxygen, one mole of lithium perchlorate is required. The molar mass of oxygen is $32\times {10}^{-3}$ kg/mole, and that of lithium perchlorate is $106.39\times {10}^{-3}$ kg/mole. Thus, the amount of lithium perchlorate required is:

$$\begin{array}{c}{m}_{LiCl{O}_4}=1.6624\left(m_{O_2,fc}+m_{O_2,prop}\right)\end{array}$$

(9)

Moreover, energy is required for the decomposition reaction of LiClO₄, and the energy consumed $E_h$ for heating LiClO₄ from ambient temperature $T_a$ to 400 °C is calculated as follows:

$$\begin{array}{c}{E}_h=m_{LiCl{O}_4}{c}_p\left(400-T_a\right),\end{array}$$

(10)

where $c_p$ is the heat capacity of LiClO₄.

3.5. Storage Tanks

Storage tanks need to be designed for the hydrogen generator to store LiH and H₂O, and the oxygen generator to store LiClO₄. However, lithium hydride, when converted to lithium hydroxide, expands to three times its original volume. Moreover, the hydrogen and oxygen generated from the hydrogen and oxygen generator, respectively, need to be stored in separate tanks before being delivered to the combustion chamber of the propulsion system. The amount of hydrogen and oxygen stored in the hydrogen and oxygen tanks is such that the robot can perform at least five hops with it. The pressure drop across the injector orifice and solenoid valve is estimated to be approximately 1 MPa, hence the hydrogen and oxygen are stored at a pressure $p_{H_2/O_2 tank}=p_c+{10}^6$ pa. The hydrogen and oxygen will be transported to the fuel-cell stack through micro pumps directly from the hydrogen and oxygen generators [41]. However, for the propulsion system, the design uses pressurized nitrogen gas to initiate the transport of hydrogen and oxygen into the combustion chamber. The nitrogen is stored at pressure $p_{N_2 tank}=2p_{H_2/O_2 tank}$. The mass of nitrogen required for transport of hydrogen and oxygen to the combustion chamber is:

$$\begin{array}{c}{m}_{N_2}=V_{H_2 tank} {\rho }_{N_2}\left(p_{H_2 tank}\right)+V_{O_2 tank} {\rho }_{N_2}\left(p_{O_2 tank}\right),\end{array}$$

(11)

where $V_{(\ast ) tank}$ is the volume of the tank storing compound $(\ast )$, and ${\rho }_{(\ast )}\left(p_{(~)}\right)$ is the density of compound $(\ast )$ at pressure $p_{(~)}$. Considering the tanks to be a spherical shell, the volume and wall thickness of all the tanks are determined from the mass and density:

$$\begin{array}{c}{V}_{(\ast )tank}=\alpha \frac{m_{(\ast )}}{{\rho }_{(\ast )}\left(p_{(~)tank}\right)}; t_{(\ast )tank}=FS\frac{p_{(\ast )tank}{r}_{(\ast )tank}}{2\sigma },\end{array}$$

(12)

where $\alpha =3$ for LiH tank, and $\alpha =1$; otherwise, $\sigma$ is the tensile strength of the material used, $r_{(\ast ) tank}$ is the radius of the tank storing compound $(\ast )$, and $FS$ is a safety factor. Subsequently, the mass of each tank is:

$$\begin{array}{c}{m}_{(\ast )tank}=4\pi r_{(\ast )tank}^2 t_{(\ast )tank} \rho ,\end{array}$$

(13)

where $\rho$ is the density of material used. Moreover, it should be noted that the mass of hydrogen and oxygen used for calculating the tank dimensions is as follows: $m_{H_2/O_2}=m_{H_2/O_2,fc}+m_{H_2/O_2,prop}$.

3.6. Other Subsystems

In addition to the power-generation and propulsion subsystems, SphereX will accommodate other subsystems such as attitude control, command and data handling, communication, instruments, power management, and external shell. The details of the attitude control, command and data handling, communication, and instruments used are provided in Appendix B. This section provides a brief description of the power management and shell subsystem.

3.6.1. Power Management

For SphereX to explore an unknown environment, a Hop+Map→Stop→Process cycle is employed, with each cycle taking 180 s. SphereX hops from its current position to a desired position, simultaneously performing the mapping of the environment with the onboard LiDAR and stereo cameras during its Hop+Map phase. After the Hop+Map phase is over, the Stop phase is initiated, where the robot stops, and then the Process phase is initiated, where the robot processes the collected data, and then the subsequent Hop+Map phase is triggered. As the robot operates in a series of different phases, each subsystem of the robot operates at different cycles, resulting in a varying power demand, as shown in Figure 6a. The life of the fuel cell is significantly reduced when connected to a varying load, due to voltage oscillations. As such, the power system is designed as a fuel cell/battery hybrid system, where the battery is constantly charged by the fuel cell, while the battery along with a power-management board handles the varying demands of the load as shown in Figure 6b.

The battery selected for our design is the GomSpace NanoPower Lithium Ion 18,650 which has a capacity of 2600 mAh @ 3.7 V, mass of 48 g, and dimensions of length 65 mm and radius 18.5 mm. With lithium-ion batteries selected for power storage, the GomSpace NanoPower P31u board is selected for power management, which is compatible with the batteries. The board offers the maximum power-point tracking (MPPT) capability and measures and logs currents, voltages, and temperatures of the power system. The board consumes only 0.165 W power under nominal conditions, weighs only 100 g, and has a dimension of 89.3 × 92.9 × 15.3 mm. However, the number of batteries required needs to be optimized. The State of Charge (SOC) of the battery is $\dot{SOC}=P_{bat}/V_{bat}Q$, where $Q$ is the nominal discharge capacity of the battery, $P_{bat}$ is the input power at any given time instant to the battery, and $V_{bat}$ is the voltage output of the battery. We can compute the power at any given time-instant as the sum of all loads as $P_{bat}=P_{fc}-P_{prop}-P_{ac}-P_{comm}-P_0$, where $P_{fc}$ is the power generated by the fuel cell system, $P_{prop}$ is the power consumed by the propulsion system, $P_{ac}$ is the power consumed by the attitude-control system, $P_{comm}$ is the power consumed by communication subsystem, and $P_0$ is the power consumed by the avionics board and scientific instruments. The battery voltage output based on SOC is:

$$V_{bat}\left(SOC\right)=\left(3+\frac{e^{SOC}-1}{e-1}\right)\left(2-e^{\lambda \frac{T-T_0}{T_0}}\right)$$

(14)

In order to avoid voltage oscillations of the fuel cell, the battery used for power storage should not experience a substantial voltage drain when connected to the load. As such, the selection of the battery is very crucial; it should have enough capacity to maintain the voltage level. The design specification is such that the battery does not operate at a voltage less than 3.5 V, which corresponds to an 85.5% state of charge. Hence, the objective of the optimization process is to minimize the discharge capacity of the battery system, such that the state of charge $SOC$ during the entire mission is greater than 85.5%, which is expressed as (18). The design variable is ${\mathsf{𝕩}}_b=Q$.

$$\min f_b\left({\mathsf{𝕩}}_b\right)=Q$$

(15a)

$$subject to: g_b\left({\mathsf{𝕩}}_b\right)\equiv SOC(\Gamma )>0.855$$

(15b)

This subsystem is also modeled as a nonlinear optimization problem (NLP), and a sequential quadratic programming (SQP) method is used to solve it. Based on the optimum discharge capacity, we calculate the number of batteries required to be $n_{bat}=ceil\left(Q/2.6\right)$ and mass to be $m_{bat}=48n_{bat}$. The batteries are stacked in columns of 4 cells inside a cuboid of size $l=0.065$, $b=0.0185\min \left(n_{bat},4\right)$, and $h=0.0185ceil\left(n_{bat}/4\right)$.

3.6.2. Shell

This discipline determines the optimal thickness of the SphereX shell for its structural rigidity upon impacting a surface with velocity $v$. An empirical approximation obtained using Zener [42] and Boettcher, Russell, and Mueller [43] measures the deformation $\delta$ (deflection of impact surface of sphere toward center of sphere) of a hollow sphere to obtain the following:

$$\delta =0.67{\left(\frac{\rho v^2}{\sigma }\right)}^{\frac{1}{2}}{\left(\frac{𝓇}{t}\right)}^{0.08}𝓇,$$

(16)

where, $𝓇$ is the radius, $t$ is the thickness of the sphere, $\rho$ is the density, and $\sigma$ is the ultimate stress of the sphere’s material. The objective function is to find the minimum thickness of the shell, such that the deformation is less than ${\delta }_{\max }$, which is a user-defined value. The design variable is the shell thickness, ${\mathsf{𝕕}}_s=t$ and the optimization problem is described as:

$$\min f_s\left({\mathsf{𝕩}}_s\right)=t$$

(17a)

$$subject to: g_s\left({\mathsf{𝕩}}_s\right)\equiv \delta <{\delta }_{max}$$

(17b)

This subsystem is also modeled as a nonlinear optimization problem (NLP), and a sequential quadratic programming (SQP) method is used to solve it. With the optimal thickness determined, the mass of the shell is determined to be $m_s=4\pi r^{2}t\rho$.

3.7. System Optimization

The objective of the system optimization process is to find the optimal mass $𝓂$ and radius $𝓇$ of the robot (SphereX) that accommodate all the subsystems while satisfying mission-specific requirements: (a) power demand, (b) mission lifetime, and (c) target-exploration distance. The task is formulated as a single-objective optimization problem with two design variables $\mathsf{𝕩}=\left[𝓂,𝓇\right]$ and two constraints. Based on the values of mass $𝓂$ and radius $𝓇$ with bounds ${𝓂}^{\left(b\right)}=\left[\begin{array}{cc}{𝓂}^{\left(L\right)}& {𝓂}^{\left(U\right)}\end{array}\right]$ and ${𝓇}^{\left(b\right)}=\left[\begin{array}{cc}{𝓇}^{\left(L\right)}& {𝓇}^{\left(U\right)}\end{array}\right]$, it is normalized between $\left[0 1\right]$ as shown:

$$\underset{\_}{𝓂}=\frac{𝓂-{𝓂}^{\left(L\right)}}{{𝓂}^{\left(U\right)}-{𝓂}^{\left(L\right)}},\underset{\_}{𝓇}=\frac{𝓇-{𝓇}^{\left(L\right)}}{{𝓇}^{\left(U\right)}-{𝓇}^{\left(L\right)}}$$

(18)

The objective function is then defined as $f\left(\mathsf{𝕩}\right)={\alpha }_1\underset{\_}{𝓂}+{\alpha }_2\underset{\_}{𝓇}$. The first constraint is defined, such that the difference between the design variable $𝓂$ and the total mass of all the subsystems ${𝓂}_{total}$ is equal to zero. The second constraint is that the assembly index is equal to one, the details of which are presented in Appendix C. The optimization process can then be formulated as:

$$\min f\left(\mathsf{𝕩}\right)={\alpha }_1\underset{\_}{𝓂}+{\alpha }_2\underset{\_}{𝓇}$$

(19a)

$$subject to: g_1\left(\mathsf{𝕩}\right)\equiv 𝓂-{𝓂}_{total}=0$$

(19b)

$$g_2\left(\mathsf{𝕩}\right)\equiv Index=1$$

(19c)

This problem is posed as a constrained optimization problem, as such the search space is divided into two regions: feasible and infeasible regions. The optimal solution found must also lie in the feasible region. As such, the nonlinear optimization problem with two nonlinear equality constraints stated above is solved using the Augmented Lagrangian Genetic Algorithm (ALGA) [44]. However, the bounds of the design variables $\mathsf{𝕩}$ are handled separately. To solve this optimization problem, the objective function and the two nonlinear constraint functions are combined using Lagrangian and penalty parameters to formulate a subproblem in the form of a cost function $\mathcal{J}\left(\mathsf{𝕩},\lambda ,𝒫\right)$ as shown in Equation (20).

$$\mathcal{J}\left(\mathsf{𝕩},\lambda ,𝒫\right)=f(\mathsf{𝕩})+{\sum }_{i=0}^2{\lambda }_i{g}_i(\mathsf{𝕩})+\frac{𝒫}{2}{\sum }_{i=0}^2{\left(g_i\right(\mathsf{𝕩}\left)\right)}^2$$

(20)

where the components ${\lambda }_i$ of the vector $\lambda$ are nonnegative and are known as the Lagrange multiplier estimates, and $𝒫$ is a positive penalty parameter.

With each subproblem solution representing a generation, a genetic algorithm is used to minimize the cost function, while making sure that the bounds are satisfied. The value of the Lagrangian multipliers and the penalty parameter are initialized at each generation, and the cost function is minimized. While minimizing each subproblem, the values of $\lambda$ and $𝒫$ are kept fixed for that particular generation. In the subsequent generations, the estimates of the Lagrangian multipliers $\lambda$ are updated, such that the subproblem is minimized within a required accuracy while satisfying the constraints. If the constraints are not met, the penalty parameter is increased by a penalty factor, which results in a new subproblem formulation, and the genetic algorithm minimizes the problem. These steps are iterated until the stopping criteria are met.

With the cost function for the constrained-optimization problem defined, the working principle of a genetic algorithm-based optimization technique to find the optimal values of the design variables is described here, as shown in Figure 7a. The algorithm starts by creating a random parent population $P_0$ of size $N_P$. Each individual of the population comprises the values of $𝓂$ and $𝓇$ chosen with a uniform distribution $𝓂=\mathcal{U}\left({𝓂}^{\left(U\right)},{𝓂}^{\left(L\right)}\right)$ and $𝓇=\mathcal{U}\left({𝓇}^{\left(U\right)},{𝓇}^{\left(L\right)}\right)$. Next, the value of the objective function $f\left(\mathsf{𝕩}\right)$ is evaluated using the subsystem discipline models as shown in Figure 7b for each individual. With the initial estimates of the Lagrange multipliers $\lambda$ and the penalty parameter $𝒫$, the fitness value is calculated for each individual according to the cost function $\mathcal{J}\left(\mathsf{𝕩},\lambda ,𝒫\right)$. After the fitness values are calculated, each individual participates in the evolution process through the genetic operators such as: selection, crossover, and mutation. This creates an offspring population $Q_0$ of size $N_Q$ [45]. For the selection operator, the tournament selection method is used. For the crossover operator, a blend crossover (BLX-α) method is used [46]. For the mutation operator, a non-uniform mutation method is used [47]. In each generation, elitism is introduced by comparing the current population with previously found best solutions. For the $t^{th}$ generation, a combined population $R_t=P_t+Q_t$ is formed of size $N_{ }+N_Q$. Next, the fitness of each individual in $R_t$ is calculated according to the cost function $\mathcal{J}\left(\mathsf{𝕩},\lambda ,𝒫\right)$. Based on the fitness value, the population is sorted in ascending order, and the best $N$ individuals are selected for the next generation. The new population $P_{t+1}$ of size $N$ is again used for selection, crossover, and mutation to create a new population $Q_{t+1}$ of size $N_Q$. The process is repeated until the desired results are obtained or the max number of generations is achieved.

4. Results

For our simulation, the genes are coded with real values of $𝓃$ and $𝓃$ with the number of populations, $N_P=50$.

Table 3. Material used for the design of the nozzle, combustion chamber, storage tanks, and robot shell.

Component	Material	$\mathbf{Tensile} \mathbf{Strength} \mathit{\sigma }$ (MPa)	$\mathbf{Density} \mathit{\rho }$ (kg/m3)
Nozzle + Combustion chamber	Stainless Steel	215	7700
Storage Tanks	Aluminum	324	2780
Shell	Carbon fiber	3500	2000

shows the materials used for the design of the nozzle, combustion chamber, storage tanks, and robot shell. Carbon fiber is the material selected for the robot shell. Given its high tensile strength and the low gravity on the lunar surface, we expect minimal challenges due to shock and vibration. Furthermore, the onboard thrusters can be used to nullify or minimize the experienced impact force to minimize risks.

4.1. Test Scenario 1: Subsurface Exploration of Mare Tranquilitatis Pit

The first simulation was run to perform the sub-surface exploration of Mare Tranquilitatis Pit on the surface of the moon at 8.33° N 33.22° E. The 10 m/pixel-resolution images from the Terrain Camera (TC) aboard the Japanese lunar orbiter Selenological and Engineering Explorer (SELENE) and 1 m/pixel-resolution images from the high-resolution Narrow-Angle Camera (NAC) onboard the Lunar Reconnaissance Orbiter Camera (LROC) revealed the dimensions of the pit. The long axis of the pit is measured to be 98 m, the short axis to be 84 m, and the depth measured from shadow measurements to be 107 m. Moreover, there is evidence that the pit opens into a sub-lunarean void of at least 20 m in extent. With the possibility of kilometers of lava-tube extension below this pit, the mission specification is defined as exploring at least 1000 m of the void. The concept of operations for performing such an exploration mission is shown in Figure 8a. A lunar lander carrying multiple SphereX robots would land near Mare Tranquilitatis Pit and deploy the robots. Each robot will undergo three phases of operation starting with 1. surface maneuvers to approach the pit; 2. pit entrance maneuver; and 3. sub-surface operation to explore the pit. Our goal is to explore 500 m on the surface in 2.5 h. In addition, it would take 10 min to enter the 50 m pit and travel 1000 m inside the lava tube in 5 h, as seen in Figure 8b.

For entering the pit, the robot needs a soft-landing maneuver, as such, in addition to the three phases discussed for the hopping controller in Appendix A, the robot has an additional soft-landing phase as shown in Figure 9a. For the soft-landing phase, the control angle $\gamma$ can be derived as $\gamma ={\tan }^{-1}\left(v_z/v_x\right)$, as shown in Figure 9b. The dynamics of the robot are:

$$\begin{array}{l}\dot{r}=v,\dot{v}=\{\begin{array}{l}g+\frac{F}{𝓂}\hfill \\ g\hfill \\ g+\frac{F}{𝓂}\hfill \end{array},\dot{𝓂}=\{\begin{array}{l}-\frac{\left|\right|F\left|\right|}{I_{sp}{g}_0} if t<t_b\hfill \\ 0 if t_b<t<t_l\hfill \\ -\frac{\left|\right|F\left|\right|}{I_{sp}{g}_0} if t_l<t<\tau \hfill \end{array}\hfill \\ \hfill \end{array}$$

(21)

The objective of the optimization process is to minimize the fuel consumption and the optimal index can be expressed as $f\left({\mathsf{𝕩}}_{sl}\right)={{\displaystyle \int }}_0^{t_b}||F||dt+{\displaystyle {\int }_{t_l}^{\tau }}||F||dt$. Four constraints are added, with the design variables ${\mathsf{𝕩}}_{sl}=\left[t_b,t_l,\tau \right]$, and the optimization problem is as follows:

$$\min f_{sl}\left({\mathsf{𝕩}}_{sl}\right)={{\displaystyle \int }}_0^{t_b}||F||dt+{{\displaystyle \int }}_{t_l}^{\tau }||F||dt$$

(22a)

$$subject to: g_{sl1}\left({\mathsf{𝕩}}_{sl}\right)\equiv t_b-t_l<0$$

(22b)

$$g_{sl2}\left({\mathsf{𝕩}}_{sl}\right)\equiv t_l-\tau <0$$

(22c)

$$g_{sl3}\left({\mathsf{𝕩}}_{sl}\right)\equiv ||r\left(\tau \right)-r_d||{}^2=0$$

(22d)

$$g_{sl4}\left({\mathsf{𝕩}}_{sl}\right)\equiv ||v_{\tau }||{}^2=0$$

(22e)

The simulation was run with the bounds on the design variables as ${𝓂}^{\left(b\right)}=\left[\begin{array}{cc}1& 8\end{array}\right]$ kg and ${𝓇}^{\left(b\right)}=\left[\begin{array}{cc}8& 30\end{array}\right]$ cm. Figure 10a shows the average cost of the population and the cost of the best/worst individual in the population. The best individual found after 40 generations is with the design variables, mass $𝓂=4.35 \mathrm{kg}$ and radius $𝓇=16 \mathrm{cm}$.

Table 4. Optimal values of relevant properties of each subsystem for SphereX.

Subsystem	Variable	Scenario 1	Scenario 2
Fuel Cell	No. of cells, 𝓃	3	3
	Current density, i	194.38 (mA/cm2)	194.38 (mA/cm2)
	Voltage of each cell, V	0.9146 (V)	0.9146 (V)
Propulsion	Combustion pressure, pc	3 (MPa)	3 (MPa)
	$\mathrm{Exit} \mathrm{pressure}, p_e$	103.57 (Pa)	309.88 (Pa)
	$\mathrm{Mixture} \mathrm{ratio}, r_m$	4.2	4.06
	$\mathrm{Contraction} \mathrm{area} \mathrm{ratio}, {ϵ}_c$	50.06	50.03
	$\mathrm{Nozzle} \mathrm{throat} \mathrm{radius}, r_{th}$	0.826 (mm)	1.2 (mm)
	$\mathrm{Nozzle} \mathrm{exit} \mathrm{radius}, r_e$	30.2 (mm)	26.5 (mm)
	$\mathrm{Nozzle} \mathrm{length} \left(\mathrm{diverging}\right), L_{n\left(div\right)}$	109.8 (mm)	94.5 (mm)
	$\mathrm{Nozzle} \mathrm{length} \left(\mathrm{converging}\right), L_{n\left(con\right)}$	4.6 (mm)	5.0 (mm)
	$\mathrm{Combustion} \mathrm{chamber} \mathrm{radius}, r_c$	5.8 (mm)	6.4 (mm)
	$\mathrm{Combustion} \mathrm{chamber} \mathrm{length}, L_c$	24 (mm)	24 (mm)
Chemicals	$\mathrm{Mass} \mathrm{of} \mathrm{LiH}, m_{LiH}$	398.2 (g)	251.1 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{LiClO}}_4, m_{LiCl{O}_4}$	745.8 (g)	452.1 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{H}}_2\mathrm{O}, m_{H_{2}O}$	872.9 (g)	560.7 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{N}}_2, m_{N_2}$	34.5 (g)	21.0 (g)
Storage Tanks	$\mathrm{Mass} \mathrm{of} \mathrm{LiH} \tan \mathrm{k}, m_{LiH tank}$	62.4 (g)	45.1 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{LiClO}}_4 \tan \mathrm{k}, m_{LiCl{O}_4 tank}$	37.8 (g)	26.5 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{H}}_2\mathrm{O} \tan \mathrm{k}, m_{H_{2}O tank}$	57.3 (g)	41.7 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{N}}_2 \tan \mathrm{k}, m_{N_2 tank}$	53.0 (g)	32.2 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{H}}_2 \tan \mathrm{k}, m_{H_2 tank}$	46.1 (g)	28.3 (g)
	$\mathrm{Mass} \mathrm{of} {\mathrm{O}}_2 \tan \mathrm{k}, m_{O_2 tank}$	12.8 (g)	7.5 (g)
Battery	Capacity, Q	280 (mAh)	198 (mAh)
	$\mathrm{No}. \mathrm{of} \mathrm{batteries}, n_{bat}$	1	1

shows optimal values of the relevant properties of each subsystem for the optimal design of SphereX. Figure 10b shows the 3D CAD model of the optimal design for the exploration of the Mare Tranquilitatis pit.

4.2. Test Scenario 2: Exploration of Victoria Crater

The second simulation was run to perform the exploration of the Victoria crater on the surface of Mars. The Victoria crater located at 2.05° S 5.50° W of the planet Mars is an impact crater and has a diameter of approximately 750 m. The Mars Exploration Rover Opportunity travelled 21 months to the Victoria crater and reached its edge on 26 September 2006, which is named “Duck Bay”. However, the rover could not explore the interiors of the crater. The concept of operations for exploring such a crater is shown in Figure 11a. A rover carrying multiple SphereXs approaches the edge of the crater and deploys them one by one. Each SphereX then enters the crater by hopping and reaches the crater base to perform experiments. Figure 11b shows the successive hopping trajectories of SphereX for exploring Victoria crater on Mars, from which the mission target is calculated as to explore 750 m inside the Victoria crater in 3.75 h.

The simulation was run with bounds on the design variables of ${𝓂}^{\left(b\right)}=\left[\begin{array}{cc}1& 8\end{array}\right]$ kg, and ${𝓇}^{\left(b\right)}=\left[\begin{array}{cc}8& 30\end{array}\right]$ cm. The best individual found after 40 generations is with the design variables mass $𝓂=3.5 \mathrm{kg}$ and radius $𝓇=15 \mathrm{cm}$. Table 4 shows the optimal values of relevant properties of each subsystem for the optimal design of SphereX.

5. Comparative Analysis

The proposed power and propulsion system can power the proposed robotic system SphereX for its mission lifetime $\Gamma$ and explore a target distance of $d_{targ}$ on a target environment using hopping mobility. We compared the proposed combined power and mobility system with other power and mobility systems in terms of its mass for exploring a target exploration distance. For all the hopping mobility systems with propulsion, the single hopping distance is restricted to $d_{hop}=10 \mathrm{m}$ for our analysis, and the robotic shell is designed such that it can maintain its structural rigidity while performing a ballistic hop of 10 m on the selected target environment. The other power systems against which our proposed power system is compared are (a) Direct Methanol Fuel Cell (DMFC), (b) Direct Borohydride Fuel Cell (DBFC), Lithium-Ion batteries, and Lithium-Polymer batteries. Two other hopping systems are compared against the proposed hopping system, which are (a) steam propulsion and (b) mechanical hopping mechanism, the details of which are provided in Appendix D.

Mass Comparison

Figure 12 shows the mass budget of SphereX for various combinations of power and the hopping system for the two test scenarios presented in Section 4. The numbers along the x-axis represents 1. FC(LiH)-H₂/O₂, 2. DMFC- H₂/O₂, 3. DBFC- H₂/O₂, 4. Li-Ion- H₂/O₂, 5. Li-Polymer- H₂/O₂, 6. FC(LiH)-Steam, 7. DMFC-Steam, 8. DBFC-Steam, 9. Li-Ion-Steam, 10. Li-Polymer-Steam, 11. FC(LiH)-Mechanical, 12. DMFC-Mechanical, 13. DBFC-Mechanical, 14. Li-Ion-Mechanical, and 15. Li-Polymer-Mechanical. It can be seen that SphereX with LiH fuel cell and H₂/O₂ propulsion is the optimal choice for both scenarios.

Figure 13 shows the power budget for the three mobility systems. It can be seen that the power requirement for SphereX with H₂/O₂ propulsion is the minimum, and that, with steam, propulsion is the maximum. Figure 13b shows that 75% of the power is required to heat the water into steam. As such, the mass of the power system for SphereX with five combinations of power and steam propulsion is the maximum, as seen in Figure 12.

Figure 14 and Figure 15 shows the mass of SphereX for various combinations of power and hopping mobility for varying exploration distances on the surface of the moon and Mars respectively. Figure 14a and Figure 15a show combinations of the five power systems with H₂/O₂ propulsion, Figure 14b and Figure 15b show that with steam propulsion, and Figure 14c and Figure 15c show that with mechanical hopping. It can be seen that, for each of the three hopping systems, the combination with the LiH fuel cell has the minimum mass. It can also be seen that, for exploration objectives less than 300 m, the combination with lithium-polymer batteries is better, but, as the exploration objectives increases, the mass of the lithium-polymer batteries increases significantly, making a LiH fuel cell the better option for long-distance exploration. For low-exploration objectives, the masses of SphereX with the three fuel cells (LiH fuel cell, DMFC, DBFC) are approximately the same, but, as the exploration objectives increase, the difference in mass is significant, making a LiH fuel cell the optimal choice. Figure 14d and Figure 15d show the comparison of the mass of SphereX for varying exploration distance with the combination of a LiH fuel cell and the three hopping options.

For this design problem, we compare the performance of Evolutionary Algorithms (EA), Particle Swarm Optimization, and Simulated Annealing [48]. Each of these approaches use stochastic techniques. Figure 16 shows the typical results we obtain for the mean of the fitness function over generations for all the three algorithms simulated and averaged over 20 times. Simulated annealing faces performance challenges and results in the premature convergence of poor solutions. Particle Swarm and Evolutionary Algorithms find feasible solutions, but, overall, we found that EA showed improved computational performance, producing solutions in half the time of PSOs, making it the better choice for this problem. Our observations also show that PSOs stagnate prematurely (more often), while EAs are more effective in continuing to innovate over the generations.

6. Conclusions

We propose to use teams of small hopping robots called SphereXs to explore extreme and rugged terrains on the moon and Mars. Here, we present an integrated power and propulsion system for SphereX containing a PEM fuel cell to generate electricity and a H₂/O₂ mobility/propulsion system in which H₂ and O₂ are stored in solid form. Lithium hydride was selected as the hydrogen source because of its high hydrogen content and the simple hydrolysis reaction required to release the hydrogen. Lithium perchlorate was selected as the oxygen source due to its high oxygen content and relatively simple catalytic decomposition reaction. The power system consists of a fuel cell-battery hybrid configuration to maximize operating life while operating at varying load conditions. The propulsion system, operating along with the attitude-control system, will provide mobility for SphereX in the form of ballistic hopping. We then formulated mathematical models for all the relevant subsystems of SphereX and applied a Multidisciplinary Design Optimization (MDO) methodology to determine an optimal design configuration of SphereX for given mission specs. To solve the MDO task, our approach utilizes a Genetic Algorithm (GA) applied at the system level integrated with multiple gradient-based optimization techniques applied at the subsystem level. This approach finds the near-optimal design configuration in terms of mass, volume, and feasibility of assembly for various missions. Our results present a near-optimal design of SphereX for two test cases: (a) the subsurface exploration of Mare Tranquilitatis on the moon, and (b) the exploration of the Victoria crater on Mars. We presented a comparative analysis of the proposed power and propulsion system with four other power systems and two other hopping systems. The comparative study showed an advantage in using our proposed system for SphereX over other design strategies to minimize the launch mass and volume.

7. Patents

The authors Himangshu Kalita and Jekan Thangavelautham report a pending patent on Spherical Robots for Off-World Surface Exploration.