How to determine the stable state of time series?

I have a year of spatial sensor data in the format unixtime x y z. The sensor is attached to a flexible strained submerged string. The string is occasionally pulled and, after some time, let go. This means that a sensor coordinate oscillates over time with different amplitudes and takes a dive on occasion.

The measurements are taken irregularly, every 1-5 minutes. However, there may be long (weeks) periods when no data is taken.

The problem here is to identify a "stable position" of the sensor - a set of coordinates that minimizes possible positioning error, assuming that the string is not being pulled.

So far, I have come to 3 ideas:

1. Quick and manual: Look at the time series, and average coordinates over the longest "stable" period.
2. Use a high-pass filter, then average the coordinates.
3. Literally optimize total distance from time series to stable point over $R^3$ (possibly add amplitude-dependent weights).

It seems like this kind of problem should arise frequently in time-series research, but I lack experience and vocabulary to define it properly. Is there a standard approach to it?

edit:

• Please note that this is not what is usually meant by a time series, because data are not obtained at regular intervals. Your question seems to imply that the true position changes after the string has been pulled, but is that really the case? – whuber Sep 13 '17 at 21:23
• I see, thank you for clarification. The true position does not change after pulling (like at the 14701.15e6 mark), but it does change within a small window over long periods of time. However, as you can see, frequency and amplitude of of oscillations does change pretty often. – monday Sep 13 '17 at 21:41
• I have attached a scatterplot for a several months of data as an example. – monday Sep 13 '17 at 21:43
• I get the impression there's a relatively slow, but real, drift in the location, assuming your vertical axis is one of the location coordinates. That suggests your problem is not one of identifying any "stable" time period--no such thing can be expected to exist--but instead it concerns smoothing the data so that you have a reasonable estimate of position at any time, an estimate that should not be overly influenced by transient large fluctuations: in other words, a robust estimate. – whuber Sep 14 '17 at 14:12