Consider a network of four receivers, two transmitters and one processor:
* the four receivers are set up such that they are not co-planar * the four receivers are fixed in position * the clocks in the receivers are not initially synchronized, but it can be assumed that they do not drift significantly * one transmitter is fixed in position; it can be assumed to be visible to all receivers can at all times * the second transmitter is mobile; it can be assumed to be visible to at least one receiver at all times * the transmitters are not synchronized with anything * the transmitters send a unique ID at regular intervals; the signal propagates at the speed of light (30 cm/ns) and is read by the receivers; there is no time information encoded in the signal, only the ID * the transmitters are "read-only" * the receivers send the transmitter ID and the receiver timestamp to the processor
If the initial configuration of the network is not known, what is the minimum setup required to determine a common time reference and the positions of the receivers?
* the processor may query a receiver to determine its current time * the processor may use the results from the fixed transmitter for any purpose * the mobile transmitter may be moved anywhere; it may or may not be visible to all the receivers * the mobile transmitter may be moved to be << 30 cm from any receiver; that is, no significant propagation delay between the transmitter and the receiver * the mobile transmitter may be moved directly underneath each receiver; it will always be visible to that receiver * one receiver may be defined as the origin * two receivers may be defined as an x-axis or a y-axis * as an absolute last resort, the positions of one or more receivers can be measured directly!
Setting up a common time reference seems easy enough, for example the Simple Network Time Protocol describes how to accurately synchronize remote clocks.
I think I can get an estimate of the z-coordinate and an orientation for each receiver by moving the transmitter next to each and then underneath. I'm stuck at determining the x-y coordinates. The restriction that the mobile transmitter is not always visible to all receivers really complicates the matter.
I'm thinking the only option is to measure the positions of all the receivers directly, that is, the "last resort" operation above.
Anyway, any assistance or references on-line or otherwise would be greatly appreciated.