# Clohessy - Wiltshere equations in electric propulsion

Further to my "asteroid chasing" question regarding the HCW equations (Clohessy - Wiltshire equations for use in Asteroid "chasing"), I was wondering if there was some transformation of these in calculations involving electric propulsion? As far as I know, these equations assume impulsive maneuvers, which is not the case for electric propulsion (although, examples from "Orbital Mechanics for Engineering Students" include a chase maneuver that takes 8 hours).

Secondly, in practical terms, since the delta V needed to perform these maneuvers are generally small in comparison to the delta V needed in other forms of trajectory design, could they be accomplished with AOCS thrusters?

• Can you explain add a link to what "my "asteroid chasing" question" and "the HCW equations" mean? Reading this question by itself, it is difficult to understand what you are asking. – uhoh Mar 21 '18 at 3:05

Practically speaking, the CW (Clohessy-Wiltshire) equations https://en.wikipedia.org/wiki/Clohessy-Wiltshire_equations are only useful for a first analytic look at understanding what is happening (they assume 2-body dynamics). For real world trajectory design these problems are solved using numeric targeting methods with full force model fidelity including finite burns.

• @uhoh. I updated my answer to be more complete. – Tom Johnson Mar 21 '18 at 18:48
• Excellent, thanks! +1 Hopefully the edit will also bump your answer back into the active list, and someone else will up vote as well! – uhoh Mar 21 '18 at 18:59
• You are absolutely correct, Tom! – ChrisR Mar 26 '18 at 14:23

I do not know at all how do you model electric propulsion, but I wonder that you assume some sort of continous thrust as:

$\ddot{x}=3n^2x+2n\dot{y}+u_x,$

$\ddot{y}=-2n\dot{x}+u_y,$

$\ddot{z}=-n^2z+u_z,$

where $u_x$, $u_y$ and $u_z$ is the continous action. In a matrix form it can be written as

$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u},$

where I assumed that $\mathbf{u}$ is the thrust and can have any time-dependency. This is a linear-time varying system which has the following analytical solution

$\mathbf{x}(t)=e^{\mathbf{A}t}\mathbf{x}_0+\int^{t}_{t_0}e^{\mathbf{A}(\Delta T - \tau)}\mathbf{B}\mathbf{u}(\tau)d\tau$,

where $\Delta T=t-t_0$ and the term $e^{\mathbf{A}t}$ is the state transition matrix (which for the HCW equations it has a closed form).

Now, the issue is to solve the integral of the control term, which depending of the assumed time evolution of $\mathbf{u}$ will have an analytical solution (e.g. if the thrust is assumed to be constant) or not. However, if an analytical solution is not possible, it can be integrated numerically.

What I wanted to remark is that HCW equations can be extended to continuous thrust models.

For your second question, you must consider that there are different types of AOCS thrusters depending on the type of operation, so the nominal situation is that you will be carrying on-board thrusters capable of providing the $\Delta V$ with the required accuracy.