Animals used in movies - how do producers ensure they're not harmed?

I was watching a re-run of the Last Samurai last night and was fascinated by all those battle scene sequences where war horses were shown being ridden at high speed by actors in full costume and made to fall as if shot or injured i.e. rolling all over, falling on top of people or clashing into other horses etc. How do they ensure that horses weren’t actually injured in a movie production? And even if they weren’t physically hurt, those stunts must be pretty stressful and scary for the animals, right? Wouldn’t the PETA folks be all over them?