Trajectory Optimization

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

Trajectory optimization is the problem of optimizing a specific performance index on the basis of trajectory design to obtain the optimal flight trajectory. Trajectory optimization formulates the trajectory design problem as an optimal control problem, solved using numerical optimization methods, to obtain the flight trajectory that is optimal with respect to a given performance index.

Core Elements

Problem Formulation

The trajectory optimization problem can be uniformly expressed as:

max⁡J=J(φ˙pr(t),A0)\max \quad J = J(\dot{\varphi}_{pr}(t), A_0)

The constraints fall into three categories:

Constraint TypeDescriptionExample
Dynamic equationsDifferential equations governing center-of-mass motionEquations of motion
Initial/terminal stateMotion states at liftoff and engine cutoffOrbit insertion elements, impact point location
Path constraintsLimitations to be satisfied during flightLoad factor, dynamic pressure, debris impact zone

Optimization Objectives

Determined by the flight mission, typical optimization objectives include:

Mission TypeOptimization Objective
Launch vehicleMaximum payload mass (equivalent to maximum residual propellant mass)
Ballistic missileMinimum impact point deviation
GeneralMinimum fuel consumption, minimum flight time

Optimization Variables

Using a GTO insertion trajectory as an example, the optimization variables include the launch azimuth and pitch program angle parameters for each stage:

U=(A0,αm,φ˙p2,t22,φ˙p31,φ˙p3h,φ˙p32,t32,…,tk32)TU = (A_0, \alpha_m, \dot{\varphi}_{p2}, t_{22}, \dot{\varphi}_{p31}, \dot{\varphi}_{p3h}, \dot{\varphi}_{p32}, t_{32}, \ldots, t_{k32})^{\mathrm{T}}

Solution Methods

Method CategoryTypical AlgorithmsCharacteristics
Shooting methodGenetic algorithms, simulated annealingGlobal search, gradient-free
Collocation methodPseudospectral methods, convex optimizationDiscretizes continuous problem
Gradient-basedSQPFast convergence, local optimum

Distinction from Trajectory Design

Comparison ItemTrajectory DesignTrajectory Optimization
ObjectiveFind a feasible solution satisfying constraintsFind the solution with optimal performance index
MethodNewton iteration, etc.SQP, genetic algorithms, etc.
Number of solutionsOne feasible solutionOptimal solution
Computational costRelatively lowRelatively high

Application Value

Trajectory optimization is an important means of fully exploiting vehicle capability. By optimizing the flight program, maximum payload capacity, minimum impact point deviation, or other optimized performance indices can be achieved while satisfying all constraints. With advances in computing power and optimization algorithms, trajectory optimization is being applied increasingly broadly in spacecraft design.

References

  • 郑伟, 安雪滢, 周祥, 何睿智. 空天飞行力学[M]. 国防科技大学, 2026.
  • 贾沛然, 陈克俊, 等. 远程火箭弹道学[M]. 国防科技大学出版社.