Mission to Schrödinger Crater
For a given desired level of science (or any other objective), what are the sets of constraints that would be acceptable?
Virtually every NASA spaceexploration mission represents a compromise between the interests of two expert, dedicated, but very different communities: scientists, who want to go quickly to the places that interest them most and spend as much time there as possible conducting sophisticated experiments, and the engineers and designers charged with maximizing the probability that a given mission will be successful (astronauts kept safe and objectives achieved) and costeffective. Recent work at JPL seeks to enhance communication between these two groups, and to help them reconcile their interests, by developing advanced modeling capabilities with which they can analyze the achievement of science goals and objectives against engineering design and operational constraints.
Our analyses conducted prior to this study have been pointdesign driven. Each analysis has been of one hypothetical case which addresses the question: Given a set of constraints, how much science can be done? But the constraints imposed by the architecture team  e.g., rover speed, time allowed for extravehicular activity (EVA), number of sites at which science experiments are to be conducted  are all in early development and carry a great deal of uncertainty. Variations can be incorporated into the analysis, and indeed that has been done in sensitivity studies designed to see which constraint variations have the greatest impact on results.
But if a very large number of variations can be analyzed all at once, producing a table that includes the majority of the trade space under consideration, then we have a tool that enables scientists and mission architects to ask the inverse question: For a given desired level of science (or any other objective), what are the sets of constraints that would be acceptable? (Note that the solution is not unique.) With this tool, mission architects could determine, for example, what combinations of rover speed, EVA duration, and other constraints produce the desired results. Further, this tool would help them identify which technologyimprovement investments would be likely to produce the largest or most important return.
However, the number of variations that need to be considered for such analysis quickly balloons to an unwieldy size. If three variations are considered for each of five constraints  a very modest example  there are a total of 243 (3 to the 5th power) variations to consider. If it takes 40 minutes to compute each variation (as when our automated optimization system, HURON, computes the complicated mission analysis described below), a full analysis consisting solely of computer runs would take 162 hours or nearly 7 days of roundtheclock computing to calculate the results. Adding further constraints or variations exponentially increases the amount of time that is needed.
In this study, we explore three methods  radial basis functions (RBF), kriging, and regression  for interpolating the bulk of the trade space based on actual computations of less than 20% of the space, which dramatically reduces the time needed to compute results over the full trade space. RBF is found to carry a higher error rate than the other two and to be the least suitable for our purposes. Choosing between kriging and regression, however, is more complicated. Depending on the intended use, one might choose kriging (as we did) for its lower average error rate or regression for its more consistent  albeit somewhat higher  error rate.
Figure 1: Geometry of a hypothetical 90day excursion from an outpost at Shackleton crater (lower right) to Schr�dinger crater and back. 
Baseline study
The subject of our study is a hypothetical mission to Schr�dinger crater, which is thought to expose underlying stratigraphic material from South Pole Aitken Basin, the oldest, largest basin on the Moon. The allotted roundtrip time from an assumed base camp at Shackleton crater to Schr�dinger and back is 90 days.
Figure 1 lays out the geometry of the mission in the baseline study. Blue dots indicate primary (required) localities for science experiments, while orange dots indicate secondary (optional) localities where scientific experiments would enhance mission results. Each of these localities comprises 6 sites where scientific activities are to be performed. The mission is conducted with two 2astronaut teams, each of which drives a pressurized rover that is periodically recharged by a separate, slower vehicle operated remotely from Earth. A target list of experiments and other activities is derived from the scientific objectives expressed by sources such as the National Research Council, the Global Exploration Strategy, and the Lunar Exploration Analysis Group. A set of constraints is provided by a missionarchitecture team. Each experiment and activity is assigned a relative science value and a cost in terms of EVA time required.
A baseline solution of the Schr�dinger excursion problem is computed, using HURON. It is found that the 90 days allowed for the mission provide ample time to conduct the desired activities at all primary sites and many secondary sites. Given these conditions, the constraint on EVA time, during which all scientific activities are conducted, is found to be the primary driver of results. A significant amount of IVA time (i.e., intravehicular activity time, when the astronauts are inside the pressurized cabins of their rovers) is included in the mission profile, during which further science activities could potentially be conducted if that were permitted and enabled.
Response surface analysis
In the next phase of the study, the objective is to determine the ranges for a variety of architecture parameters that achieve equivalent levels of science return. With 3 variables for each of 5 constraints, the trade space is an irregular, multidimensional (aka hyperdimensional) grid. Fortytwo cases are run on HURON, and the remaining 83% of cases are interpolated using each of the three methods stated above.
The analysis produces results for 5 parameters: productivity, cost, mission duration, kilometers traversed, and percentage of targeted experiments conducted (in which each experiment is weighted by the importance assigned by scientists participating in the study). We choose to sort the results by this percentage, but the sorting could just as easily be done by any or all of the other parameters. Results are validated and an error rate is computed for each interpolation method (Figure 2).
Figure 2: Comparison of the error rates of 3 interpolation methods. 
Kriging (the purple curve) is found to have the lowest average error rate and is used to compute a table in which the 243 cases are ranked by the percentage described above (Table 1). The rows near the top of the table thus represent the cases that would likely be selected by mission architects who wish to maximize the return from experiments conducted during this mission. Given this information, they would be able to further winnow the choices according to other constraints that are not included in this analysis  e.g., public outreach considerations or participation by other countries.
Input Set ID 
EVA Constraint (h/day) 
Locality Duration (hours) 
Rover Speed (km/h) 
Number of Localities (required/optional) 
Egr/Ingr Time (mins) 
% Targeted Experiments Conducted 
Source 
235 
5 
5 
15 
17 / 35 
10 
104.2 
Estimated 
154 
4 
5 
15 
17 / 35 
10 
103.6 
Estimated 
226 
5 
5 
10 
17 / 35 
10 
103.2 
Estimated 
73 
3 
5 
15 
17 / 35 
10 
103.0 
Estimated 
145 
4 
5 
10 
17 / 35 
10 
102.6 
Estimated 

215 
5 
4 
15 
12 / 27 
20 
77.1 
Computed 
134 
4 
4 
15 
12 / 27 
20 
76.9 
Computed 
147 
4 
5 
10 
17 / 35 
30 
76.6 
Estimated 
53 
3 
4 
15 
12 / 27 
20 
76.2 
Estimated 
66 
3 
5 
10 
17 / 35 
30 
76.0 
Estimated 
206 
5 
4 
10 
12 / 27 
20 
75.9 
Computed 
125 
4 
4 
10 
12 / 27 
20 
75.7 
Computed 
182 
5 
3 
15 
17 / 35 
20 
75.1 
Estimated 
187 
5 
3 
15 
12 / 27 
10 
75.1 
Estimated 
44 
3 
4 
10 
12 / 27 
20 
75.0 
Estimated 

45 
3 
4 
10 
12 / 27 
30 
56.7 
Estimated 
31 
3 
4 
5 
7 / 13 
10 
56.5 
Estimated 
239 
5 
5 
15 
7 / 13 
20 
56.0 
Estimated 
198 
5 
4 
5 
12 / 27 
30 
55.8 
Estimated 
158 
4 
5 
15 
7 / 13 
20 
55.4 
Estimated 
230 
5 
5 
10 
7 / 13 
20 
55.3 
Estimated 
117 
4 
4 
5 
12 / 27 
30 
55.0 
Estimated 
77 
3 
5 
15 
7 / 13 
20 
54.7 
Estimated 
149 
4 
5 
10 
7 / 13 
20 
54.7 
Estimated 
221 
5 
5 
5 
7 / 13 
20 
54.6 
Estimated 

168 
5 
3 
5 
7 / 13 
30 
22.7 
Estimated 
96 
4 
3 
10 
7 / 13 
30 
22.4 
Estimated 
87 
4 
3 
5 
7 / 13 
30 
21.9 
Estimated 
15 
3 
3 
10 
7 / 13 
30 
21.8 
Computed 
6 
3 
3 
5 
7 / 13 
30 
21.2 
Computed 
Table 1. The product of the response surface analysis. Sections are shown from the top, middle and bottom of the table, which consists of 243 cases. The orangeheaded columns are the 5 constraints discussed above. The blueheaded column gives the number by which the cases are ranked: the percentage of targeted experiments that are conducted, in which each experiment is weighted by the importance assigned by scientists participating in the study.
Conclusions
We have shown that we can survey a very broad range of combinations of architect parameters through a combination of computer optimization runs and interpolation, and that we can provide an estimate of the quality of the results by a number of different methods. Increasing the number of computer runs decreases the interpolation error rate. Of the three interpolation methods employed in this study, kriging (which had the lowest average error rate) was found to be best for our purposes, but the more consistent results of the regression method (the yellow curve in Figure 2) may be preferable in certain circumstances.
For more information, contact Charles Weisbin at
Charles.R.Weisbin@jpl.nasa.gov.