Final-Year Project: Gradient-Based Orbital Transfer Optimisation with MCMC-Driven Random Walkers

An orbital dynamics simulation that models the docking manoeuvre of two spacecraft using differential equations, optimises fuel usage and transfer time with gradient descent, and estimates uncertainty using MCMC-based random walkers.

---

📚 Table of Contents

• 🧠 Project Motivation
• 📐 Technologies Used
• 🔭 Project Scope
• ⚙️ How It Works
• 📊 Visualisations
• 🚧 Challenges
• 🎯 Intended Use
• 📌 Limitations
• 🤝 Credits

---

🧠 Project Motivation

This project emerged from my final-year research at the University of Hertfordshire, where I aimed to apply theoretical astrophysics to a practical engineering problem—docking optimisation in multi-body orbital mechanics. The goal was to simulate and improve autonomous spacecraft docking via algorithmic optimisation and probabilistic modelling.

---

📐 Technologies Used

• Python
• numpy, scipy – numerical methods, ODE solvers
• matplotlib, seaborn – plotting and animation
• emcee – Markov Chain Monte Carlo for uncertainty quantification
• PyMC – Bayesian regression (optional extension)
• LaTeX – scientific documentation and reporting

---

🔭 Project Scope

This simulation models:

• Orbital parameters (e.g. positions & velocities in x, y & z-axis) of an interceptor spacecraft
• Optimal velocity (v) and trajectory angle (θ) using gradient descent
• Cost function J(v, θ) minimised iteratively
• Uncertainty quantification via Bayesian inference using MCMC
• Animated visualisation of convergence and trajectory paths

---

⚙️ How It Works

1. Define physical model: orbital mechanics via Newtonian physics
2. Formulate cost function: based on docking constraints
3. Optimise: gradient descent updates (v, θ) iteratively
4. Quantify uncertainty: use emcee to explore the posterior distribution
5. Visualise: plot contours, convergence, and trajectory path

🔬 See my Jupyter notebook for a step-by-step walkthrough.

---

📊 Visualisations

• Descent path over cost surface
• Convergence of cost function
• Posterior distributions of burn parameters
• Optimised Trajectory

Fig. 1: Gradient descent over the contour surface plot of the cost function J. The white dashed line on the left shows the cost of a classic Hohmann transfer.

Fig. 2: Changes in the cost function J over gradient descent steps. Notice that it is not a perfectly horizontal line after reaching the valley (around 130 steps) of the cost function. The descent process becomes much slower when the minimum is close.

Fig. 3: Simulation of random walkers to search for the final converged parameters (v, θ). We observe a sign of stable convergence at around 150 iterations (red dashed lines) for both parameters.

Fig. 4: Results of the optimised transfer. Note that a heavier penalty (grey dashed line '---') allows for faster transfer.

---

🚧 Challenges

• Dealing with stiffness in ODE solvers
• Tuning learning rate and momentum in gradient descent
• Ensuring MCMC convergence and burn-in
• High computational time for repeated sampling

---

🎯 Intended Use

• Academic: share with peers, mentors, and for viva demonstration
• Public: open-source learning resource for aerospace and AI students
• Portfolio: showcase numerical optimisation and simulation capabilities

---

📌 Limitations

• Assumes a 2D docking plane and a simplified gravitational model
• Ignores fuel leakage, actuator delay, or collision dynamics
• MCMC assumes Gaussian noise

---

🤝 Credits

• Tszon Tseng – project author
• Supervised by Dr. James Collett and Professor Detlef Mueller
• Inspired by orbital mechanics applications & gradient-based ML methods in Deep Learning
• MCMC logic adapted from emcee GitHub

---

📬 Contact

Feel free to reach out via LinkedIn or raise an issue if you'd like to collaborate or suggest improvements.

---