Oblivious Machines and Input Dependency
According to the WP article on Turing machine equivalents, oblivious machine head position is independent of input and is a function of time instead. To wit:


  An oblivious Turing machine is a Turing machine where movement of the various heads are [sic] fixed functions of time, independent of the input. In other words, there is a predetermined sequence in which the various tapes are scanned, advanced, and written to.


REFERENCES

https://dl.acm.org/citation.cfm?doid=322123.322138

SEE ALSO

Head Position Function of Oblivious Turing Machines
https://stackoverflow.com/questions/14847080/how-does-an-oblivious-turing-machine-work