Clohessy-Wiltshire (CW) Equation

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

Definition

The Clohessy-Wiltshire (CW) equation, also known as the Hill-Clohessy-Wiltshire (HCW) equation, is a set of linearized dynamical equations describing the relative motion of a chaser spacecraft with respect to a target spacecraft in a near-circular reference orbit. Expressed in the target's Local Vertical Local Horizontal (LVLH) coordinate frame, the equations decompose relative motion into radial, along-track, and cross-track components.

The equations originate from Hill's (1878) work on Earth-Moon relative motion and were applied to spacecraft rendezvous by Clohessy and Wiltshire (1960), becoming a cornerstone of relative orbital dynamics.

Mathematical Form

In the LVLH frame, with relative position $(x, y, z)$ and velocity $(\dot{x}, \dot{y}, \dot{z})$ , and reference orbit mean motion $n = \sqrt{\mu / a^3}$ , the CW equations in matrix form are:

\begin{pmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} \end{pmatrix} = \begin{pmatrix} 3n^2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -n^2 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} + \begin{pmatrix} 0 & 2n & 0 \\ -2n & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix}

Expanded scalar form:

Radial (x): $\ddot{x} - 3n^2 x - 2n\dot{y} = 0$
Along-track (y): $\ddot{y} + 2n\dot{x} = 0$
Cross-track (z): $\ddot{z} + n^2 z = 0$

Assumptions

The derivation of the CW equations relies on:

Near-circular reference orbit: The target spacecraft is in a circular or near-circular orbit (eccentricity $e \approx 0$ )
Small relative distance: The separation between chaser and target is much smaller than the reference orbit radius ( $\rho \ll a$ )
Two-body gravity field: Only the central body's gravity is considered; perturbations (drag, SRP, third-body) are neglected
Linearization: The nonlinear relative equations are Taylor-expanded to first order, dropping higher-order terms

Due to the linearization, CW equations are accurate only when $\rho / a < 0.01$ . For large-scale relative motion in cislunar space, more precise nonlinear models are required.

Analytical Solution

The CW equations admit closed-form analytical solutions. Given initial state $(x_0, y_0, z_0, \dot{x}_0, \dot{y}_0, \dot{z}_0)$ :

x(t) = \frac{\dot{x}_0}{n}\sin(nt) - \left(\frac{2\dot{y}_0}{n} + 3x_0\right)\cos(nt) + \left(\frac{2\dot{y}_0}{n} + 4x_0\right)

y(t) = \frac{2\dot{x}_0}{n}\cos(nt) + 2\left(\frac{2\dot{y}_0}{n} + 3x_0\right)\sin(nt) - (3\dot{y}_0 + 6nx_0)t + \left(y_0 - \frac{2\dot{x}_0}{n}\right)

z(t) = \frac{\dot{z}_0}{n}\sin(nt) + z_0\cos(nt)

The cross-track motion (z) is an independent harmonic oscillation, decoupled from the in-plane motion. The in-plane motion (x-y plane) contains periodic terms and a secular drift term (proportional to $t$ ), meaning uncontrolled relative motion is generally unstable.

Applications

CW equations are widely used in spacecraft engineering:

Rendezvous and docking: Designing transfer trajectories from far-range approach to final docking, foundational for space station logistics
Formation flying: Designing relative orbit configurations and control strategies for satellite formations
Proximity operations: Relative motion planning for debris removal, on-orbit servicing
Spacecraft intention recognition: Comparing observed relative motion data against CW-predicted trajectory patterns to infer noncooperative target motion intentions (e.g., approach, flyby, rendezvous)
Collision risk assessment: Propagating covelliances to compute collision probabilities

Relation to CR3BP

Both CW equations and the Circular Restricted Three-Body Problem (CR3BP) describe relative motion, but they apply in different regimes:

Feature	CW Equations	CR3BP
Gravitational bodies	Two-body (central body + reference spacecraft)	Three-body (two primaries + massless particle)
Orbit type	Relative motion near a circular orbit	Periodic/quasi-periodic orbits near libration points
Linearization	Yes	Nonlinear (can be linearized near equilibrium points)
Typical applications	Rendezvous, formation flying	DRO, NRHO, Halo orbit design

In cislunar missions, CW equations are used for relative motion analysis near space stations (e.g., Tianzhou cargo spacecraft rendezvous with the space station), while CR3BP is used for larger-scale orbit design (e.g., DRO formations, NRHO missions).

References

Clohessy W H, Wiltshire R S. Terminal guidance system for satellite rendezvous[J]. Journal of the Aerospace Sciences, 1960, 27(9): 653-658.
Hill G W. Researches in the lunar theory[J]. American Journal of Mathematics, 1878, 1(1): 5-26.
Curtis H D. Orbital Mechanics for Engineering Students[M]. 4th ed. Butterworth-Heinemann, 2020.
Jing H, Sun Q, Dang Z, Wang H. Intention Recognition of Space Noncooperative Targets Using Large Language Models. Space Sci. Technol. 2025;5:0271.