optimal_planetary_landing_with_constraints

This is the Python CasADi implementation code for the article titled "Optimal planetary landing with pointing and glide-slope constraints".

Optimal Control for Constrained Planetary Landing 🚀

This repository contains the numerical implementation and theoretical analysis of a vertical powered descent problem for planetary landing. The project models the optimal trajectory of a lander while strictly adhering to glide-slope (state) constraints and thrust pointing (input) constraints.

The continuous optimal control problem is discretized and solved using Python and CasADi/IPOPT. The implementation numerically verifies the theoretical derivations from the source paper, successfully reproducing the "Max-Min-Max" (Bang-Bang) optimal control structure.

📄 Reference Article

This project is based on the theoretical framework and proofs established in the following research paper:

Title: [Optimal planetary landing with pointing and glide-slope constraints]
Authors: [Clara Leparoux, Bruno Hérissé, Frédéric Jean]
Link: Read the full article here

⚙️ System Dynamics

The vehicle's motion is modeled using point-mass dynamics in a two-dimensional inertial frame.The state vector includes position $(r)$, velocity $(v)$, and mass $(m)$. The continuous-time differential equations governing the flight are:

• Kinematics: $\dot{r} = v$
• Kinetics: $\dot{v} = \frac{T}{m}u - g$ (where (T) is maximum thrust, (u) is the normalized thrust vector, and (g) is gravity)
• Mass Depletion: $\dot{m} = -q ||u||$ (where $q$ is the maximum mass flow rate)

📊 Simulation Results

The numerical optimization confirms the mathematical proofs. The solver successfully finds a trajectory that strictly respects the physical boundaries (the $5^\circ$ glide-slope cone and $45^\circ$ pointing limits) while maximizing remaining propellant.

Simulation Parameters & Initial Conditions (Constant Mass)

• Initial Position: $(x_0, z_0) = (2000 \text{ m}, 1500 \text{ m})$
• Initial Velocity: $(v_{x0}, v_{z0}) = (100 \text{ m/s}, -75 \text{ m/s})$
• Initial Mass ($m_0$): $1905 \text{ kg}$
• Maximum Thrust ($T$): $16573 \text{ N}$
• Thrust Bounds: $0.3 \le ||u|| \le 0.8$
• Gravity ($g_0$): $3.71 \text{ m/s}^2$ (Mars)
• Mass Flow Rate ($q$): $0 \text{ kg/s}$

Figure: The simulated optimal trajectory (x vs. z), the bang-bang control norm profile (Max-Min-Max), and the saturated thrust pointing direction.

🛠️ Dependency

• CasADi (Nonlinear Optimization Framework)
• IPOPT (Interior Point Optimizer)
• Matplotlib (Visualization)