I asked a similar question the other day, but this is broader and clearer. Is there ever a circumstance where an animal companion will do something or can be controlled without a trick besides riding it?
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You can use Handle Animal to push an animal to do something it doesn't know a trick for.
You can use the Speak With Animals spell to communicate and ask the animal for a favor. This isn't magical control, so you will have to persuade them. Druids can use Wild Empathy to make the animal more friendly.
In the case of an animal companion it's already friendly provide it's not being mistreated, so unless the request is obviously suicidal or counter to the animal's nature, it likely won't be difficult to make such a request successful.
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You can also use Charm Animal to do the same thing with more forceful magical persuasion, or Dominate Animal for even more forceful magical persuasion. On your own companion, neither of those should be necessary.