Hey Michael,
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if you touch or click the red/yellow “Emergency Stop” button that is displayed on upper right of the display, then yes, it is programmed to stop it. The controller enters “estopped mode” then.
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If you hit the big red hardware button that is on top of the Onefinity Controller,
it will not enter the “estopped mode” (at least not if the manufacturer did not change its wiring), because then it just cuts power only to the controller, as explained here. However in this case it may be that if ‘tool-enable’ is set to ‘lo-hi’, the positive signal on pin 15 of the 25-pin I/O port breaks down immediately which would lead to the IoT relay having no trigger signal anymore and therefore may shut the router off.
You can eliminate this deficiency by connecting a big red button to pin 23 “estop” of the 25-pin I/O port, as explained here. If you press it then, it will put the controller into “estopped” mode, which will take care to stop the router by disabling ‘tool-enable’ or the spindle via ModBus.