Trajectory Equation

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The trajectory equation is a system of differential equations describing the center-of-mass motion of a spacecraft during engine operation. The trajectory equation projects thrust, aerodynamic forces, gravity, and control forces onto a selected coordinate system in a unified manner and solves for the spacecraft's velocity, position, and attitude as functions of time through numerical integration.

Core Elements

Basic Composition

The trajectory equation consists of the following components:

Component	Content	Description
Dynamic equations	Newton's second law projected onto each axis	Core equations
Kinematic equations	Differential relationships between velocity and position	$\dot{x} = v\cos\theta$ , etc.
Control equations	Attitude control commands	$\delta_\varphi = a_0^\varphi(\varphi - \varphi_{pr})$
Supplementary equations	Auxiliary computations for mass, altitude, angle of attack, etc.	Closed-form equation system

Form in the Launch Frame

In the launch frame, the trajectory equation includes 6 dynamic equations (3 for center of mass + 3 for rotation about center of mass), along with kinematic and supplementary equations. The center-of-mass dynamic equation is:

m\frac{d^2\mathbf{r}}{dt^2} = \mathbf{P} + \mathbf{R} + \mathbf{F}_c + m\mathbf{g} + \mathbf{F}_k'

Projected onto the three axes of the launch frame, this yields a system of scalar equations suitable for numerical integration.

Form in the Velocity Frame

Selecting the velocity frame as the computation frame allows the trajectory equation to be separated into longitudinal and lateral equations of motion:

Equation Type	Key Parameters	Described Motion
Longitudinal equations	$v$ , $\theta$ , $x$ , $y$	Motion in the firing plane
Lateral equations	$\sigma$ , $z$	Motion perpendicular to the firing plane

Cutoff-Point Parameters

The trajectory equation is integrated up to the cutoff time $t_k$ (fuel depletion or commanded shutdown) to obtain the cutoff-point parameters $(v_k, \theta_k, x_k, y_k)$ . These parameters determine the subsequent trajectory of the passive phase.

Application Value

The trajectory equation is the core tool for spacecraft design and mission analysis. By solving the trajectory equation, the flight trajectory can be predicted, cutoff-point parameters can be determined, and flight performance can be analyzed. For ballistic missiles, the cutoff-point parameters determine impact accuracy; for launch vehicles, they determine insertion accuracy. The trajectory equation also serves as the foundation for trajectory optimization and guidance algorithm design.

References

Zheng W, An X Y, Zhou X, He R Z. Aerospace Flight Mechanics[M]. National University of Defense Technology, 2026.
Jia P R, Chen K J, et al. Long-Range Rocket Ballistics[M]. National University of Defense Technology Press.