A troubleshooting checklist only helps you identify problems you've already seen, or at least anticipated. For a 3+ year old automobile, following the manufacturer's checklist, you'll be covered 99.44+% of the time. For a robot that you started designing in January, not so much.
Not to say that you shouldn't have a checklist; you should. Include checks for things you've already seen, and things you might expect (as described by several posters above). This checklist may be able to find 75% to 90% of your problems, or even a bit more, depending on how thoroughly you made it. Part of this checklist may be to run the robot in "test mode". For this to work, you have to program a test mode which will provide very simple open loop inputs to each actuator so that you can verify that the output signal results in the appropriate behavior. If you use feedback sensors, the results of these sensors should be reported as well.
Once through the checklist, follow a method. Your individual techniques may be a bit more scientific or a bit more engineering-based depending on your predispositions, but the key points are:
- Describe what is going wrong (in terms of behavior, not cause).
- Run a test case or two with a few variables different to verify that behavior.
- Adjust your description if necessary.
- Form a hypothesis (or several) as to what is causing the problem.
- Test the hypotheses - look for a test which distinguishes among them (preferably which cuts the list in half or smaller based on the results). Get down to one thing, or at least a small set.
- Repair/replace/bypass/probe that one (small set of) thing.
- Repeat steps as necessary, eliminating one or more possible causes each pass.
- When you get it working, look for an update your troubleshooting checklist that would identify the issues you had or thought you might have.