Velocity Frame

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

Definition

The velocity frame is a coordinate system established with respect to the direction of the spacecraft's flight velocity, denoted as $o_1 - x_v y_v z_v$ and abbreviated as $V$ . The origin of the velocity frame is at the spacecraft's center of mass, and the $x_v$ axis is aligned with the flight velocity direction. It is used to describe the velocity vector state of the spacecraft and the decomposition of aerodynamic forces.

Core Elements

Axis Definitions

Axis	Definition
$o_1 x_v$	Along the flight velocity direction of the spacecraft
$o_1 y_v$	In the principal plane of symmetry, perpendicular to the $x_v$ axis
$o_1 z_v$	Perpendicular to the $x_v o_1 y_v$ plane, pointing to the right when viewed in the direction of flight

Relationship with the Body Frame

The relationship between the velocity frame and the body frame is defined by two angles:

Angle	Symbol	Definition
Angle of attack	$\alpha$	Angle between the projection of the velocity vector in the principal plane of symmetry and the $x_1$ axis
Sideslip angle	$\beta$	Angle between the velocity vector and the principal plane of symmetry

The direction cosine matrix is:

\mathbf{V}_B = \mathbf{R}_x(-\beta) \cdot \mathbf{R}_y(\alpha)

Relationship with the Launch Frame

The relationship between the velocity frame and the launch frame is defined by three angles:

Angle	Symbol	Definition
Flight path yaw angle	$\sigma$	Angle between the projection of the velocity vector in the horizontal plane of the launch frame and the $x$ axis
Flight path angle	$\theta$	Angle between the velocity vector and the horizontal plane
Bank angle	$\gamma_v$	Rotation angle about the velocity vector

Aerodynamic Force Decomposition

In the velocity frame, aerodynamic forces naturally decompose into:

Component	Direction	Physical Meaning
Drag $X$	$-x_v$ direction	Opposing the velocity direction
Lift $Y$	$y_v$ direction	Perpendicular to velocity, in the principal plane of symmetry
Side force $Z$	$z_v$ direction	Perpendicular to velocity and the principal plane of symmetry

Application Value

The velocity frame is the core coordinate system for aerodynamic force analysis and flight mechanics. Aerodynamic forces (lift, drag, side force) naturally decompose along the velocity frame, and the angle of attack and sideslip angle are key parameters for aerodynamic calculations. During powered-phase trajectory design, the velocity frame is used to establish the equations of motion for the velocity vector. For the reentry phase of cislunar missions, the velocity frame provides the foundation for analyzing aerodynamic heating and deceleration.

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.