How do I extract specific information from the /diagnostics topic?

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First, have you read the diagnostics wiki ? It states :

The diagnostics toolchain is built around the /diagnostics topic. On this topic, hardware drivers and devices publish diagnostic_msgs/DiagnosticArray messages with the device names, status and specific data points.

If you are able to visualize the battery state with rqt_robot_monitor then you can access to the battery state from the /diagnostics topic. This topic receives diagnostic_msgs/DiagnosticArray which is a header and a list of diagnostic_msgs/DiagnosticStatus messages (message definition here).

If you want to monitor the battery state you can susbscribe to the /diagnostics topic, then in the list of the components in the status attribute of the message search for the one which name is Battery (or if it's different write the same name as in rqt_robot_monitor). Once you have found it you can monitor the battery state as desired.

I think the diagnostic aggregator can be useful for you.