Orbital Rendezvous LQI / MPC

AstroScale GNC home-assignment submission by Yuval Grimberg.

This repository contains a reproducible orbital mechanics and rendezvous-control study:

• Q1: two-body orbit propagation with an intentional Earth-impact trajectory and a corrected collision-avoidance trajectory.
• Q2: Clohessy-Wiltshire rendezvous control with cached LQI, deterministic noise support, custom initial conditions, and an experimental Kalman-estimated constrained MPC comparison.
• Streamlit viewer: live Q1/Q2 simulations, browser-native animations, custom Q2 initial conditions, and controller notes.

Highlights

• Preserves the submitted assignment structure: Q1-Yuval_Grimberg.py, Q2-Yuval_Grimberg.py, and the original report PDF.
• Runs without Bayesian optimization by default by using the cached report Q matrix.
• Terminates Satellite B at Earth impact instead of propagating through Earth.
• Adds a constrained MPC comparison that respects the same +/- 1 N thrust limits used by the LQI simulation.
• Includes a Streamlit app for visual review and parameter exploration.
• Includes LaTeX/Tectonic-rendered controller equations under docs/equations/.

Visual Results

Q1 summary	Q2 cached LQI nominal

Q2 animation	LQI vs Kalman MPC

Quick Start

Python >= 3.9 is required.

python -m pip install -r requirements.txt

Run the full validation:

python validate_submission.py

Launch the live app:

streamlit run streamlit_app.py

Then open:

http://localhost:8501

Repository Layout

README.md                              GitHub-facing runbook
LICENSE                                MIT license
requirements.txt                       Python dependencies
Q1-Yuval_Grimberg.py                   Q1 two-body orbit propagator
Q2-Yuval_Grimberg.py                   Q2 CW LQI / MPC rendezvous simulation
propagator.py                          Compatibility wrapper for Q1
rendezvous.py                          Compatibility wrapper for Q2
streamlit_app.py                       Live Q1/Q2 browser viewer
validate_submission.py                 Lightweight reproducibility gate
docs/README.md                         Documentation index
docs/assets/                           README images and GIFs
docs/control/                          LQI vs Kalman-MPC notes
docs/equations/                        LaTeX equation source and compiled PDF
docs/report/                           Submitted report PDF
docs/submission/                       Archive, live-run, and report-fix notes
outputs/                               Generated plots, GIFs, and metrics

The root stays focused on runnable entry points; supporting notes and report artifacts live under docs/.

Q1: Earth Impact And Avoidance

Satellite A is a circular 700 km altitude reference orbit. Satellite B uses eccentricity e = 0.999 with the same semi-major axis and is intentionally an Earth-impact case. Satellite C is the corrected avoidance trajectory with the same apogee as B and perigee equal to A's orbital radius.

The two-body model is:

$$ \ddot{\mathbf r}=-\frac{\mu}{|\mathbf r|^3}\mathbf r,\qquad \mu=G M_E $$

Satellite B is stopped by an event function:

$$ g(t)=|\mathbf r_B(t)|-R_E $$

with terminal downward crossing. The script reports impact time, impact radius, impact position, minimum radii, and the delta-V needed to move from B's apogee speed to C's apogee speed.

Run Q1:

python Q1-Yuval_Grimberg.py --save
python Q1-Yuval_Grimberg.py --animate --save
python Q1-Yuval_Grimberg.py --smoke --save

Outputs:

outputs/q1/metrics.json
outputs/q1/q1_summary.png
outputs/q1/q1_animation.gif

Q2: CW Rendezvous Control

Q2 uses the Clohessy-Wiltshire linearized relative dynamics around the 700 km circular reference orbit:

$$ \begin{aligned} \ddot{x}-3n^2x-2n\dot{y} &= \frac{F_x}{m},\\ \ddot{y}+2n\dot{x} &= \frac{F_y}{m},\\ \ddot{z}+n^2z &= \frac{F_z}{m}, \end{aligned} \qquad n=\sqrt{\frac{\mu}{r_0^3}}. $$

The cached LQI baseline solves the continuous Riccati problem on the augmented state:

$$ \dot{z}_I=Cx,\qquad J=\int_0^\infty \left( \begin{bmatrix}x\z_I\end{bmatrix}^{T} Q_{\mathrm{aug}} \begin{bmatrix}x\z_I\end{bmatrix} +u^TRu \right)dt. $$

The raw LQI command is saturated to the actuator limit:

$$ u=\mathrm{clip}(-K_xx-K_Iz_I,-1,1)\ \mathrm{N}. $$

Run the cached controller:

python Q2-Yuval_Grimberg.py --save
python Q2-Yuval_Grimberg.py --smoke --save
python Q2-Yuval_Grimberg.py --t-final 60000 --save
python Q2-Yuval_Grimberg.py --animate --save

Use deterministic measurement noise:

python Q2-Yuval_Grimberg.py --noise --save --seed 1

Run a custom initial condition in LVLH coordinates [x y z vx vy vz], with meters and meters/second:

python Q2-Yuval_Grimberg.py --initial-condition 120 -80 40 0.02 -0.03 0.01 --scenario-name "Custom IC" --save

Cached LQI vs Kalman-Estimated MPC

The experimental MPC comparison uses the same CW model but discretizes it:

$$ x_{k+1}=A_dx_k+B_du_k $$

and solves a bounded finite-horizon problem:

$$ \begin{aligned} \min_{u_0,\ldots,u_{N-1}} \quad & \sum_{k=0}^{N-1}x_k^TQx_k+u_k^TRu_k+x_N^TPx_N\\ \text{s.t.}\quad &x_{k+1}=A_dx_k+B_du_k,\\ &-1\leq u_{x,k},u_{y,k},u_{z,k}\leq 1. \end{aligned} $$

With noise enabled, MPC receives a Kalman-filter state estimate rather than the raw noisy measurement:

$$ \hat{x}^{-}_{k+1}=A_d\hat{x}_k+B_du_k,\qquad \hat{x}_{k+1}=\hat{x}^{-}_{k+1}+K_{k+1}(y_{k+1}-\hat{x}^{-}_{k+1}). $$

Run the comparison:

python Q2-Yuval_Grimberg.py --smoke --noise --compare-mpc --save

Run the optional profile selector using the mission-oriented right-cost score:

python Q2-Yuval_Grimberg.py --t-final 300 --dt 10 --noise --compare-mpc --optimize-mpc --no-save --mpc-dt 60 --mpc-horizon 6

The right-cost score is:

$$ S=d_f+250v_f+0.01J_u+2p_{\mathrm{sat}}+0.05t_{\mathrm{settle}} +\mathbb{1}_{\mathrm{not\ settled}}0.25t_f. $$

Lower is better. It rewards small final miss distance, low closing speed, lower effort, less saturation, and faster capture.

Saved comparison metrics:

outputs/q2/mpc_comparison_metrics.json

Equation Sheet

The core Q2 equations are also maintained as LaTeX and compiled with Tectonic:

docs/equations/q2_controller_equations.tex
docs/equations/build/q2_controller_equations.pdf

Preview:

To regenerate with a local Tectonic install:

tectonic --outdir docs/equations/build docs/equations/q2_controller_equations.tex

Streamlit Viewer

The Streamlit app is the easiest way to inspect behavior interactively:

• Q1 ECI animation and LVLH paths.
• Q2 cached LQI animation and plots.
• Q2 custom initial conditions.
• Cached LQI vs Kalman MPC comparison.
• Controller notes and README in-app.

streamlit run streamlit_app.py

Expected Outputs

outputs/q1/metrics.json
outputs/q1/q1_summary.png
outputs/q1/q1_animation.gif
outputs/q2/metrics.json
outputs/q2/mpc_comparison_metrics.json
outputs/q2/*_distance.png
outputs/q2/*_lvlh_trajectory.png
outputs/q2/*_states_controls.png
outputs/q2/1_nominal_scenario_animation.gif

Assumptions And Limits

• Q1 uses a two-body Earth model only: no drag, perturbations, or non-spherical gravity.
• Satellite B is intentionally an impact trajectory.
• Q2 uses linearized CW dynamics around the 700 km circular reference orbit.
• Thrust saturation is +/- 1 N per axis.
• The cached LQI controller remains the default submission controller.
• The Kalman-MPC controller is an experimental comparison path, not a flight-ready controller.
• A production-quality controller would need sensor geometry, actuator dynamics, keep-out constraints, collision constraints, and Monte Carlo validation.