# Clohessy-Wiltshire equations vs. Lambert solver for low-Earth Orbital Rendezvous

Just a quick question: Is a Lambert solver a generalization of the Clohessy-Wiltshire equations when regarding, for example, orbital rendezvous around Earth?

• I can't answer right now, so just a comment: No. They are very different things. Lambert's problem is a boundary value problem. The CW equations are a linearization of the equations of motion in the vicinity of a spacecraft, expressed in the (rotating) LVLH frame that spacecraft. – David Hammen Jan 22 '15 at 21:47

They are very different things.

A Lambert solver is one way to bring a spacecraft close enough to another spacecraft where that first spacecraft can use the Clohessy-Wiltshire (CW) equations to guide the way through the rendezvous process. The chaser (the active spacecraft in the rendezvous) can't use CW to make the transition from far away / far below to near field rendezvous. On the flip side, the chaser can't use Lambert's technique to guide the final rendezvous. The two techniques address two very different problems in two very different ways.

Lambert's problem addresses how to transfer from point $A$ at time $t_A$ to point $B$ at time $t_B$ subject to the influence of a gravitational field, plus some means to change velocity to accomplish the transfer. The CW equations don't address this problem.

The CW equations are instead a linearization of the rotating frame equations of motion in the vicinity of a spacecraft, where the rotating frame is the target vehicle's Local Vertical / Local Horizontal (LVLH) frame. The linearization makes for easily computable second derivatives, which makes it very useful in onboard flight software. While this approximation is fairly accurate at close distances, it's terrible at large distances.

Update: Answers to two questions raised in a comment.

Is there a reason why a Lambert solver can't be used for the final part of the rendezvous?

There's not just one reason. There are many reasons.

A Lambert solver yields a burn-coast-burn transfer, where both burns are impulsive. This isn't how proximity operations work. Prox ops instead involves a (potentially large) number of tiny burns that keep the vehicle on a pre-planned trajectory to the docking port (or to the capture box in the case of HTV, Dragon, and Cygnus). The pre-planned trajectory is designed in part with those numerous small burns in mind. A Lambert solver may well yield a solution that is mind numbingly bad in terms of delta V.

Even more important is the primary reason for that pre-planned trajectory: It keeps things safe. A trajectory that intersects with the target along the way from point A to point B is a monumentally bad idea. A Lambert solver doesn't address the nature of the space that intervenes from point A to point B; that space is implicitly assumed to be empty space. That assumption doesn't apply in the vicinity of the target vehicle. Lambert cannot be used during prox ops for this reason alone.

Yet another reason is pluming. Rocket exhaust inevitably contains a small portion of unburnt fuel, unburnt oxidizer, or both. Pluming the Space Station with hydrazine and/or nitric acid is another one of those monumentally bad ideas. Lambert solutions don't address where the exhaust goes; it presumably goes into empty space. That assumption fails during proximity operations. A Lambert solution not only can but will plume the target, particularly on final approach.

Yet another reason is computation. Lambert's problem is a boundary value problem; these are typically computationally expensive to solve. Lambert's problem is not an exception to this rule. This is not a problem when the chaser is far away from the target. Simply enable the Lambert solver tens of seconds (or whatever it takes) before the solution is needed, thereby giving the solver time to find the solution. This doesn't work during prox ops, where algorithms need to be fast and timely.

Numerical precision is another computational issue. Lambert solvers work in Earth centered inertial coordinates. CW-based techniques work in local LVLH coordinates. That is by far the preferred frame for proximity operations. Going from local LVLH to ECI and back to local LVLH sacrifices over half of the precision of a double precision number. With single precision numbers, the precision loss is nearly 100%.

All in all, using a Lambert solver during proximity operations is worse than monumentally bad.

Are the CW equations used by real spacecraft rendezvous systems, or is something more advanced utilized?

CW is the dominant approach used to rendezvous with the Space Station. (I'm tempted to say it's the only approach used for ISS rendezvous.) The linear nature of the equations make them fast and make them fit very well with a Kalman filter.

The Clohessy-Wiltshire equations make two key explicit assumptions: The small angle assumption and a circular target orbit assumption. The small angle assumption means that CW becomes less accurate with increasing distance. That's not a problem. You don't use CW at large distances. They're used for near field rendezvous and proximity operations. The second assumption isn't a problem with the Space Station. It's eccentricity is typically less than 0.003, which is extremely low. The assumption of a circular orbit is a problem for a formation of satellites in an elliptical orbit. Nonetheless, a CW-like technique is still used. There are a number of techniques that yield equations that look very much like the CW equations for elliptical orbits.

• Is there a reason why a Lambert solver can't be used for the final part of the rendezvous? Also, are the CW equations used by real spacecraft rendezvous systems, or is something more advanced utilized (i.e. more accurate equations)? – InquisitiveInquirer Jan 24 '15 at 7:19
• @user7388 - I've updated my answer to address those two questions. – David Hammen Jan 24 '15 at 19:51