Orbital Rendezvous Control Project

Advanced Control & Systems

Luenberger-Based Rendezvous Control with Angle-Only Data

Academic Project | Control Systems Theory & Simulation

1. Context & Engineering Problem

Orbital rendezvous (docking) is an essential maneuver requiring precise control of a satellite's 4 relative states (radial/angular position and velocity) to bring them to zero. The central engineering challenge is an **output feedback problem**:

Full State Required: The controller needs all 4 states: $x = [r, \dot{r}, \theta, \dot{\theta}]^T$.
Limited Measurement: The sensors only provide a single measurement: the angle $y = \theta$.
Unobservable States: All velocities ($\dot{r}, \dot{\theta}$) and the radial position ($r$) are unknown.

Critical Discovery: Initial analysis revealed the physical model (using meters, seconds) was numerically ill-conditioned, producing physically unrealizable controller gains of $10^{15}$. This problem was solved via **system normalization (non-dimensionalization)** before design.

2. Goal

Designed a complete orbital rendezvous autopilot capable of guiding a chaser satellite to a target using only angle-of-sight data ($y=\theta$). The core challenge was the non-observability of the system in Cartesian coordinates, which I solved using a coordinate transformation to modified polar coordinates for the Luenberger Observer.

3. Project Milestones

MILESTONE 1

Feasibility Analysis

Proved the system is both Controllable and Observable from $y=\theta$. The problem is solvable.

Achieved

MILESTONE 2

Robust Gain Design

Calculated stable $K_{norm}$ and $L_{norm}$ gains after normalization (Observer 10x faster than Controller).

Achieved

MILESTONE 3

Observer Validation

Simulation proved estimation error $\tilde{x}(t) \to 0$. The observer successfully tracks the real state.

Achieved

MILESTONE 4

Full System Validation

Simulation proved the satellite's real state $x(t) \to 0$. The rendezvous maneuver was successful.

Achieved

4. Simulation Results Gallery

Real vs Estimated States — Validation of Observer (Milestone 3): Estimated states (dotted) rapidly converge on real states (solid).

Estimation Error vs Time — Observer Performance: All four estimation errors converge to zero, confirming successful state tracking.

Real Satellite State vs Time — Full System Validation (Milestone 4): The satellite's real states $x(t)$ are all driven to zero.

Rendezvous Trajectory Phase Plot — Rendezvous Trajectory: The phase plot shows the satellite spiraling into the origin (0,0), confirming a successful rendezvous.

Control Effort vs Time — Control Effort (Thrusters): Shows the commanded force $u(t)$ converging to zero as the rendezvous is completed.

5. Conclusion

The project was a complete success. We demonstrated that an autopilot for orbital rendezvous can be designed using only a single angular measurement. The analysis revealed that **system normalization** was a critical, non-negotiable step to overcome numerical instability and achieve a robust, physically realistic design. Final validation confirmed the (Observer + Controller) architecture successfully stabilizes the satellite and achieves rendezvous.

6. Future Work (Logical Next Steps)

This report serves as a proof-of-concept for the 2D linear model (4 states). The logical future work is to extend this solution to the full, non-linear, 6-Degree-of-Freedom (6-DOF) problem.

13-State Modeling: Utilize the full state vector (3D position, 3D velocity, 4 attitude quaternions, 3 angular velocities).
Non-Linear Estimator: Replace the linear Luenberger Observer with an Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF) to fuse data from multiple sensors (e.g., Camera + LiDAR).
6-DOF Control: Design a control law that manages both the 3 translation forces ($\vec{F}$) and the 3 attitude torques ($\vec{\tau}$) of the spacecraft.