Asteroid Mining Tech Demo: Trajectory Design Analysis

📌 Abstract

This repository contains the solution for Assessed Exercise 3 of the Programming and Mathematics for Astrodynamics and Trajectory Design (2024-2025) module.

The project involves a grid search for asteroid sample return opportunities for commercial nanosatellite demonstrators. The mission targets Near-Earth Asteroid (NEA) mining, specifically identifying Asteroid 2014 WX202 as the optimal candidate. Transfer trajectories between early 2033 and the end of 2037 were analyzed to meet strict hyperbolic excess velocity and rendezvous maneuver constraints.

🎯 Mission Requirements

• Launch Window: Jan 1, 2033 – Dec 31, 2037.
• Mission Profile: Earth $\to$ Asteroid (Stay 2-6 months) $\to$ Earth.
• Constraints:
  • Departure/Arrival $v_{\infty} < 1.5$ km/s.
  • Rendezvous maneuvers $\Delta v < 500$ m/s.

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🏆 Results

The analysis identified Asteroid 2014 WX202 as the prime candidate for the sample return mission. The optimal mission profile requires a Total $\Delta v$ of 5.477 km/s and spans a total duration of 560 days (approx. 1.5 years).

Mission Event	Date (MJD2000)	Date (Gregorian)	Time (days)	$\Delta v$ (km/s)
Earth Departure	12363.5	05 Nov 2033	0	1.391
Asteroid Arrival	12673.5	11 Sep 2034	310	0.299
Scientific Operations	—	—	60	0.000
Asteroid Departure	12733.5	10 Nov 2034	—	0.541
Earth Return	12923.5	19 May 2035	190	3.247

Porkchop Plots

Below are the porkchop plots generated for the four distinct burns of the mission.

Leg 1.1: Earth Departure and Asteroid Arrival

Leg 1.2: Asteroid Rendezvous Maneuver

Leg 2.1: Asteroid Departure Maneuver

Leg 2.2: Earth Return and Arrival

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⚙️ Methodology

Due to the dataset size (~20,000 asteroids), a brute-force Lambert search was infeasible. A multi-step pruning process was implemented to filter candidates down to a manageable number before performing high-fidelity analysis.

Process Overview

The selection strategy involved an initial pruning using the Shoemaker-Helin approximation, followed by a custom Figure of Merit (FoM) filter, and finally a rigorous Lambert arc scan.

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📂 Repository Structure

.
├── main.m                  # Main script for pruning and delta-v calculations
├── Leg1ConditionsPar.m     # Parallel computation for Leg 1 (Earth -> Asteroid)
├── Leg2ConditionsPar.m     # Parallel computation for Leg 2 (Asteroid -> Earth)
├── ShoemakerHelin.m        # Initial asteroid pruning based on delta-v approximation
├── results.mat             # Saved final calculation results
│
├── lambert/                # Directory for Lambert's problem solvers
│
├── plot/                   # Folder with final plots and plotting functions
│
├── class_excercises/       # Foundational exercises from class
│
├── ex1/                    # Solution for a previous assignment (Earth-Mars transfer)
│
└── README.md               # This file

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🧮 Appendix: Mathematical Proof

The Pruning Figure of Merit

To efficiently select the top 100 candidates, a custom Figure of Merit (FoM) was derived. This metric approximates the $\Delta v$ required to match the asteroid's orbit based on its eccentricity ($e$) and inclination ($i$), assuming low inclination/eccentricity and a circular Earth orbit.

I. Change Eccentricity ($\Delta v_1$)

The change in velocity required is the difference between the periapsis velocity ($v_p$) and the circular velocity ($v_c$).

$$ \Delta v_1 = |v_p - v_c| $$

Using the vis-viva equation where $r = a$, the circular velocity is:

$$ v_c = \sqrt{\frac{\mu}{a}} $$

The periapsis velocity at $r_p = a(1-e)$ is:

$$ v_p = \sqrt{\frac{\mu}{a}} \cdot \sqrt{\frac{1+e}{1-e}} $$

Using a 1st order Taylor approximation for $e < 0.2$:

$$ \sqrt{\frac{1+e}{1-e}} \approx 1+e $$

Substituting this back yields:

$$ \Delta v_1 \approx \left| \sqrt{\frac{\mu}{a}} \cdot (1+e) - \sqrt{\frac{\mu}{a}} \right| = v_c \cdot e $$

II. Change of Inclination ($\Delta v_2$)

With no change of velocity magnitude (preserving semi-major axis $a$):

$$ \Delta v_2 = 2 v_c \cdot \sin(\Delta i / 2) $$

III. Final Figure of Merit

Since the two maneuvers are applied perpendicularly ($\Delta v_1$ in-plane, $\Delta v_2 \perp$ plane), the total $\Delta v$ is:

$$ \Delta v_{tot} = \sqrt{(v_c \cdot e)^2 + \left( v_c \cdot [2 \cdot \sin(i/2)] \right)^2} $$

Normalizing by $1/v_c$ gives the final FoM used in the code:

$$ \boxed{FoM = \sqrt{e^2 + [2 \cdot \sin(i/2)]^2}} $$