Mueller was hired by Kreiselgeräte Company in 1933. He developed the PIGA accelerometer. and worked on gyroscopes for the German Navy. Later on, he worked on the guidance and control system for the A3 test rocket, the A5, and the A4 (V-2) ballistic missile.
Under Project Paperclip, Mueller emigrated to the United States on 16 November 1945 with the Argentina group. There, he worked on developing guidance systems for the PGM-11 Redstone, PGM-19 Jupiter, MGM-31 Pershing, and the Saturn I missiles. In 1960 Mueller left NASA for private industry. 
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- Roger E. Bilstein (1999). Stages to Saturn: A Technological History of the Apollo/Saturn Launch Vehicle. DIANE Publishing. ISBN 978-0-7881-8186-3.
- Michael J. Neufeld (2008). Von Braun: Dreamer of Space, Engineer of War. Random House, Inc. ISBN 978-0-307-38937-4.