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For an overview of various models of power armor in the Fallout series of games, see Power armor.
FOBoSLogoThe following is based on Fallout: Brotherhood of Steel and has not been confirmed by canon sources.

The advanced power armor in Fallout: Brotherhood of Steel is not like the power armors used by the Brotherhood of Steel, or Enclave, or American military forces before the Great War. It does resemble the T-51b power armor a bit, however. The origin of this armor is probably an attempt by Vault-Tec to produce some valuable combat equipment to help their own Secret Vault to successfully enter the post-apocalyptic world. The advanced version was probably made in order to improve the first prototype, but was never fully implemented.

Vault-Tec had access to vast knowledge of the government and other companies and probably tried to use all the technologies they could get access to. Having no option to manufacture or buy T-51b power armor in big amounts, they probably started their own research and produced something that was at least similar to the original.

The only one such known advanced power armor could be found in Secret Vault laboratories.

Other valuable armors like Metal and Tesla, that could be found in FO:BOS, might be produced by Vault-Tec as well during the same development project.