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Similarly to

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which mission used the first multiprocessor in space?

The Saturn V rocket control computer used three processors executing the same program to increase reliability by redundancy.

This question is about the first distributed computing in space, several independent digital processors executing different programs within a single space probe. General purpose processors with a memory for data and program.

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    $\begingroup$ You'll need to define processor. Early vehicles had a lot of modules dedicated to processing specific analog or digital signals, and also sequencers, which may not be what you intended. $\endgroup$
    – DrSheldon
    Jul 14 at 23:17
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The first space probe using distributed computing was Voyager:

Voyager employs three dual-redundant computer systems per spacecraft. The first, the CCS, is nearly identical to that flown on Viking, performing sequencing and spacecraft health functions along with new ones necessitated by the addition of the other computers. Telemetry data formatting and transmission handled by the Flight Data System are done on Voyager with the help of a custom-built computer. Attitude control and articulation of the scan platform are accomplished with the third computer system. One concept from the STAR computer proposed for the TOPS, applicable to Voyager, is longer dormancy. if unpowered JPL's pro B ect . Although staff believed both CCSs that equipment are always would powered, last rarely are both Flight Data Systems running, and both attitude control computers are never turned on at the same time. Full bit-for-bit redun- dancy is not maintained in the dual memories. For example, "expended" algorithms, such as the deployment sequence executed shortly after separation from the booster, need not be maintained5. Both memories are accessed by the single active processor in each system. The Flight Data System keeps a copy of its instructions in both memories, but intermediate data and variables can be stored in either memory. This seemingly casual attitude toward memory duplication tightens up considerably near encounter periods, which is one time that both CCS processors are in tandem mode.

Source: Computers in Spaceflight, p. 174

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