Adding external encoder to reduce backlash of linear track

Depends. The 7th axis being mathematically coupled is going to make this much harder. Are you trying to achieve realtime coordinated motion of all 7 axes? Or could you get away with just using E1 as a high-precision positioner separate from the rest of the robot?

If E1 were asynchronous, you could control it from RSI, and simply feed the desired position from the Robot Interpreter to RSI when you need E1 to move. If that motion has to be synchronized with A1-A6... for each motion, you would need to pre-calculate a TCP distance, then live-track the TCP's progress over that distance while "dangling a carrot" in front of the RSI module controlling E1, varying the distance based on how fast the TCP is moving. And if the TCP path is circular or non-linear in some other way....

Honestly, the simplest method to improve the accuracy of a linear axis is to eliminate backlash. Simply avoiding any reversals, or applying anti-backlash motions to keep the preload of the gearing constantly biased in a consistent direction, gives great improvements for minimal investment.

As posted above, a lot of your issues might be navigated by taking a different programming approach, your solution does sound like adding an extreme amount of complexity that might be avoided with simple solutions,

That said there are other things you could consider such as laser tracking and correction, but without knowing the project or available budget its hard to advise.

Quote from bats

Skyfire can you elaborate on
"Simply avoiding any reversals, or applying anti-backlash motions to keep the preload of the gearing constantly biased in a consistent direction, gives great improvements for minimal investment."

For an asynchronous 7th axis, the simplest anti-backlash procedure is to always end a motion in the same direction. That is, if moving from -100 to +100, nothing else is needed. But if moving from +100 to -100, the axis should be moved to, say, -110, then to -100. The idea is that, whenever moving the axis to a position where accuracy is required, the final motion should always be in the + direction. For a simple linear axis, this eliminates almost 100% of any backlash error. Basically, backlash only becomes a problem when the axis reverses direction.

If you are performing calibration of the axis, this rule also applies. As long as you use the same anti-backlash motions during calibration as during production, your accuracy should improve greatly.

This also applies to the full robot arm. By using such techniques in my programs (in either joint space or Cartesian), I was able to greatly improve accuracy for aerospace fuselage assembly.

For a fully synchronous 7th axis, this becomes more difficult, but it can be programmed. Break your program into sections that only need the 7th axis to move in the + direction, and only move the axis in the negative in subroutines added in between the accuracy-critical sections. Or use a simple anti-backlash motion on the 7th axis at the beginning of each accuracy-critical program. The trick is to only allow reversal of the axis during parts of the program where accuracy is non-critical, and to re-establish backlash correction at the beginning of any accuracy-critical path. And during the accuracy-critical path, the axis must only move in one direction, or remain stationary, never reverse.

A simplistic example:

PTP StartPoint ; non-accurate move, all axes
PTP_REL {E1 -10} ; first half of anti-backlash move
PTP_REL {E1 10} ; second half of anti-backlash move
PTP EndPoint ; as long as EndPoint is in a + direction on E1, no backlash errors of E1 should have any effect