A Mathematical Model for Interplanetary Logistics
This article demonstrates a methodology for designing and evaluating the operational planning for interplanetary exploration missions. A primary question for space exploration mission design is how to best design the logistics required to sustain the exploration initiative. Using terrestrial logistics modeling tools that have been extended to encompass the dynamics and requirements of space transportation, an architectural decision method has been created. The model presented in this article is capable of analyzing a variety of mission scenarios over an extended period of time with the goal of defining interesting mission architectures that enable space logistics. This model can be utilized to evaluate different logistics trades, such as a possible establishment of a push-pull boundary, which can aid in commodity pre-positioning. The model is demonstrated on an Apollo-style mission to both provide an example and validate the methodology.
The development of an interplanetary supply chain requires the unification of two traditionally separate communities: aerospace engineering and operations research. In order to create an effective means of communication between both communities, a distinct terminology has been developed and is detailed extensively in Section I. Specifically, the definition of the commodities or supplies, and the elements or physical containment and propulsion units used to transport the commodities are detailed. Furthermore, the network definition is presented as well as the definition and description of the time expanded network, which is the terrestrial modeling technique employed for the space logistics model. Section II describes the components of the interplanetary logistics problems. Section III presents the problem formulation and constraints. In Section IV a description of the optimization methodology developed to solve this problem is discussed. In Section V the problem formulation and solution methodology is applied for the example of an Apollo-style mission to both explain the implementation and validate the methodology presented. Section VI reviews the contributions of this article and describes continuing work in this area.
The goal of the interplanetary logistics problem is to determine feasible mission architectures to satisfy the demand generated by the needs of exploration. The key concept of the interplanetary logistics problem is that the demand of crew, consumables, equipment and other exploration requirements at in-space locations drives the mission requirements. Therefore, the first required input for the interplanetary logistics problem is the definition of these supplies. For example, if the exploration mission is a sortie style mission to investigate a particular location, the demand might consist of a few crew members at a specific location and the supplies necessary to both support the crew and enable the exploration activities.
Given the demand of the mission, it is necessary to determine how and when the supplies on Earth will be transported to the in-space locations. As missions become more complex and evolve over a period of time, a solution may become less obvious. Since the goal is to minimize the cost of any mission, it is desirable to optimize the timing and method of transport of the supplies to in-space locations. Therefore, it is necessary to define all pathways and structures used for transport, and allow the optimizer to analyze the different architectures to select the best one.
Given this information, the interplanetary logistics problem can determine low cost mission architectures that satisfy the exploration demand. The solution generated will detail the scheduling and assignment of supplies to vehicles for in-space transport and launch scheduling requirements. More importantly, however, the output of this problem can be used to determine a push-pull boundary for the supplies, the potential of a specific location, either on a surface or in-space for storing supplies, benefits of in-situ resource utilization over multiple missions, or even the sensitivity of mission architectures to changes in vehicle parameters.
The first step in developing a model for interplanetary logistics is defining a concrete nomenclature that describes the components of the problem. The problem fundamentally consists of three components: the commodities or supplies that must be shipped to satisfy a mission demand, the elements or physical structures used to both hold and move the commodities, and the network or pathways the elements and commodities travel on. The following sub-sections define the parameters that describe each of these components.
The goal of the space logistics project is to determine how to meet the demand for the exploration missions. As such, we are investigating how to optimally ship multiple types of commodities. For the purpose of the logistics problem, a commodity will be defined as a high-level aggregate of a type of supply, such as crew provisions. Thus, we will define a set of k = 1,…, K commodities, each with the following parameters:
* Denote the demand of each commodity as d^sup k^.
* Denote the origin of each commodity as so^sup k^.
* Define the destination of each commodity as sd^sup k^.
* Define the availability interval of each commodity as to^sup k^ = [sto^sup k^, eto^sup k^], where sto^sup k^ is the starting time of the interval and eto^sup k^ is the ending time of the interval.
* Define the delivery interval of each commodity as td^sup k^ = [std^sup k^, etd^sup k^], where std^sup k^ is the starting time of the interval and etd^sup k^ is the ending time of the interval.
* Define the unit mass of each commodity as m^sup k^ when it arrives at the destination.
* Define the unit volume of each commodity as v^sup k^ when it arrives at the destination.
* Define the number of specified waiting sequences as nw^sup k^.
By defining a waiting sequence as part of the commodity input, a number of wait arcs along the path can be specified, which allows onroute destinations to be designated. For each waiting arc sequence I where 0
* Define the static node of the wait sequence as sw^sub l^^sup k^.
* Define the required waiting time period as pw^sub l^^sup k^.
* Define the wait interval for each wait sequence as = [tw^sub l^^sup k^, etw^sub l^^sup k^], where stw^sub l^^sup k^ is the starting time of wait interval l of commodity k, etw^sub l^^sup k^ is the ending time of wait interval l of commodity k, and etw^sub l^^sup k^ - stw^sub l^^sup k^ ? pw^sup l^^sup k^.
It is important to note that in this model a crew member is treated as a commodity. In practice crewed missions are treated differently during mission planning: however, for the purposes of the architectural design tool created by this model, crew can be considered a commodity with highly restrictive parameter values. By narrowing the availability and delivery windows for a crew commodity, the feasible shipment pathways are limited and reasonable architectures for crewed flights can be obtained.
Elements
In order to ship the commodities from the origin to the destination locations, we require ‘containers’ to both hold the commodities and provide propulsion to move the mass through space. These components can be abstracted to a single definition of an element. Elements are physical, indivisible functional units that transport the commodities from origin to destination. An element is classified by the amount of commodity capacity and propulsive capability it possesses. Elements can be divided into two classes: non-propulsive elements M^sub N^ and propulsive elements Mp. The element parameters are (Figure 1) as follows:
* The maximum fuel mass of a propulsive element m, m euro M^sub p^ is denoted by mf^sup m^.
* The specific impulse of the fuel in element m is denoted by I^sub sp^^sup m^.
* The structural mass of element m is denoted by ms^sup m^.
* The mass capacity of element m is denoted by CM^sup m^.
* The volume capacity of element m is denoted by CV^sup m^.
* The cost of element m is denoted by Cost^sup m^.
Networks
In order to transfer the commodities and elements from the origin node to the destination node, the trajectories must be defined. The purpose of the interplanetary logistics model developed in this article is to analyze the multiple choices available for routing all of the commodities and elements to determine the best logistics architecture. To model the different available trajectories, a network model of space is created to represent the possibilities available for transferring commodities to their respective destination. The following sections detail the development of the space network utilized to form the model presented in this article.
The physical network, or static network, represents the set of physical locations, or nodes, and the connections, or arcs, between them. The physical nodes, or static nodes, represent the different physical destinations in space, including the origin and destination of all the commodities, as well as the possible locations for transshipment. Three types of nodes have been identified: Body nodes, Orbit nodes and Lagrange point nodes. These classifications distinguish the type of information required to define a node of each type. The physical arcs, or static arcs, represent the physical connections between two nodes, that is, an element can physically traverse between these two nodes. We define an arc (si, sj) to be a static arc that represents a feasible transfer from static node si to static node sj.
The mathematical description of the static network is given below:
* Define the static network as a graph GS, where GS = (NS, AS).
* Define the set of nodes, NS = {s1,…, sn}, in the static network.
* Define the set of arcs, AS ? NS × NS in the static network.
An example of an Earth-Moon static network is provided in Figure 2. In this picture we can see the connection of the Earth surface nodes to the Earth orbit node, representing launches and returns. Similarly, the lunar surface nodes are connected to the lunar orbit node, representing descent and ascent trajectories. In addition, the orbit nodes, as well as the first Earth-Moon Lagrangian point, are connected by in-space trajectories.
In order to analyze sequences of missions that evolve over an extended period of time, and to account for the time-varying properties that can arise in certain astrodynamic relationships, we have chosen to introduce time expanded networks as a modeling tool. In the time expanded network the absolute time interval under consideration is discretized into T time periods of length ?t. A copy of each static node is made for each of the time points and the nodes are connected by arcs according to the following rules:
* The arc must exist in the static network.
* The arc must create a connection that moves forward in time.
* The arc must represent a feasible transfer, with respect to the orbital dynamics.
The mathematical description of the time expanded network is given below:
* Define the time expanded network as a graph G, where G = (N, A).
* Define the set of nodes in the time expanded network as N = {i = (si, t) si euro NS, t = 1,…,T}. To simplify the notation, for a given node i euro N, let s(i) and t(i) denote the physical node and the time period corresponding to node i, i.e., if i = (si, t) then s(i)= si and t(i)= t.
* Define node s as the general source that generates the supply of elements. This node is connected to every node in the network where an element can originate.
* Define the set of arcs in the time expanded network as A ? N x N. An arc a = (i, j) = ((si, t), (sj, t + T^sup t^^sub si,sj^)) exists if and only if there exists an arc (si, sj) in the static network, and the transit time from static node si to static node sj starting at time t is T^sup t^^sub si,sj^. Note that if si = sj, then T^sup t^^sub si,sj^ =1 for all t.
* Define path p as a sequence of nodes. In particular, let f(p) and l(p) denote the first node and the last node of path p. If path p originates at node s, f(p) = s for all such p.
Using the static network depicted in Figure 2, we can create the time expanded network in Figure 3. Here, the time expanded network is notional as not all arcs are represented, but how the trajectories evolve in time can be readily seen.
To account for the fact that on certain transfer arcs two burns occur, we slightly modify the time expanded network. We first introduce a new fictitious static node labeled fie. Note that this node is not related to the static network. On every transfer arc (i, j), s(i) ? s(j) requiring two burns we add a new auxiliary node k = (fic, t) with two arcs; one connects i to k and the other one k to j. The value of t is irrelevant. In this new network, each arc (i, j) with s(i) ? s(j) corresponds to a single burn. All such arcs are called burn arcs and we denote the set of all burn arcs as AB.
The fuel mass fraction, which represents the ratio of the fuel mass to the initial mass, for element m to execute the burn corresponding to arc a C A^sub 8^ is defined as:
The execution of a space mission requires logistical decisions at every step. Logistics are required to accumulate all of the required commodities for space missions, as well as procure and assemble all elements at the launch site. However, since at the time of launch all of the items required to perform a space mission are co-located at the launch pad, the terrestrial logistics can be decoupled from the interplanetary logistics model. Therefore, the interplanetary logistics model encompasses all of the logistical decisions required between the launch pad and the locations in-space.
There are numerous decisions made during space missions that can be modeled and optimized to create a better mission description. Although, from a system perspective, it would be desirable to make all of these decisions concurrently, due to computational limitations this is not a reasonable approach. Instead, the interplanetary logistics model is decomposed into three fundamental components: launch packing and scheduling, element packing and in-space network optimization.
Launch is a highly constrained transportation activity, where although traditional allocation and packing decisions are required, many additional constraints are necessary to model a feasible launch. For this reason the launch problem is decoupled at Low Earth Orbit (LEO), creating a boundary between the launch allocation and the in-space network optimization. This assumption is assumed to be only slightly restrictive, since for many mission architectures there exists a delay at LEO before proceeding to in-space destinations. Launching focuses on selecting the appropriate elements to perform the launch, satisfying the payload requirements for launch, and scheduling requirements for launch vehicles and launch sites.