Trajectory Planning

Author: CislunarSpace
Site: https://cislunarspace.cn

Definition

Trajectory planning designs optimal or near-optimal flight paths from start to target for stratospheric airships under given mission constraints. For long-endurance regional station-keeping missions, trajectory planning must simultaneously consider energy efficiency, station-keeping precision, and flight time.

Problem Formulation

State Space

\mathbf{x} = [x, y, z, \dot{x}, \dot{y}, \dot{z}, m]^T

Control Input

\mathbf{u} = [T_x, T_y, T_z, \dot{m}_{ballast}]^T

Objective Function

J = \int_{t_0}^{t_f} L(\mathbf{x}, \mathbf{u}, t) dt + \phi(\mathbf{x}(t_f))

Typical objectives:

Objective	Expression
Minimum energy	$\min \int P_{prop} dt$
Minimum time	$\min (t_f - t_0)$
Minimum path length	$\min \int
Maximum station-keeping precision	$\min \int

Constraint Conditions

Path Constraints

Constraint	Requirement
Station-keeping region	$\mathbf{p} \in \mathcal{R}_{station}$
Maximum range	$d_{total} \leq D_{max}$
No-fly zone	$\mathbf{p} \notin \mathcal{R}_{forbidden}$

Typical Algorithms

Intelligent Optimization

Method	Characteristics
Particle Swarm Optimization (PSO)	Global search
Genetic Algorithm (GA)	Strong robustness
Deep Reinforcement Learning (DRL)	Handles complex constraints

Low-energy Trajectory

Principle

Utilize wind field energy to reduce propulsion consumption:

E_{prop} = \int \frac{1}{2}\rho | \mathbf{v}_{rel} |^3 C_D S \cdot dt

Optimal Strategy

Strategy	Condition
Downwind navigation	$\mathbf{v}_{wind}$ pointing toward target
Altitude scheduling	Utilizing vertical wind shear
Wait/drift	During no-fly zones or low-wind periods

References

Betts J T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming[M]. SIAM, 2023.
Zhang Y, et al. Trajectory Optimization for High Altitude Airship[J]. AIAA Journal of Aerospace Systems, 2024.