Longitudinal and Lateral Motion

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

Definition

In the velocity coordinate frame, through the small-angle assumption and the instantaneous balance assumption, the spatial equations of motion can be decomposed into two independent sets of equations: the longitudinal motion equations describe motion within the firing plane (the $xoy$ plane of the launch coordinate frame), and the lateral motion equations describe motion perpendicular to the firing plane. This decomposition achieves decoupling of the three-dimensional motion, greatly simplifying trajectory analysis.

Core Elements

Longitudinal Motion Equations

Longitudinal motion describes the vehicle's motion within the firing plane, with the following principal parameters:

Parameter	Symbol	Description
Velocity magnitude	$v$	Flight speed scalar
Flight path angle	$\theta$	Angle between the velocity vector and the horizontal plane
Horizontal position	$x$	Component along the launch frame $x$ -axis
Vertical position	$y$	Component along the launch frame $y$ -axis

The longitudinal motion equations are:

\begin{cases} m\dot{v} = P_e - C_x q S_M + mg\sin\theta \\ mv\dot{\theta} = (P_e + C_y^\alpha q S_M)\alpha + mg\cos\theta \\ \dot{x} = v\cos\theta \\ \dot{y} = v\sin\theta \end{cases}

Lateral Motion Equations

Lateral motion describes the vehicle's deviation from the firing plane, with the following principal parameters:

Parameter	Symbol	Description
Heading error angle	$\sigma$	Angle of velocity vector deviation from the firing plane
Lateral position	$z$	Component along the launch frame $z$ -axis

Lateral motion is typically small and is maintained near the firing plane by the lateral control system.

Decoupling Conditions

The conditions for decoupling longitudinal and lateral motion:

Small-angle assumption ( $\alpha$ , $\beta$ , $\sigma$ , $\nu$ are small quantities)
Instantaneous balance assumption
Neglecting Earth rotation effects (for preliminary calculations)

In precise calculations, longitudinal and lateral motion are coupled and must be solved simultaneously.

Application Value

The decomposition of longitudinal and lateral motion is the foundational method for powered-phase trajectory analysis. Longitudinal motion determines the vehicle's speed, flight altitude, and range, forming the core of trajectory design. Lateral motion reflects the degree of deviation from the firing plane and must be suppressed by the lateral control system. During the preliminary design phase, the longitudinal trajectory is typically designed first, followed by analysis of lateral motion stability.

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.