In this paper we prove decidability results of restricted fragments of simultaneous rigid reachability or SRR, that is the nonsymmetrical form of simultaneous rigid E-unification or SREU. The absence of symmetry forces us to use different methods, from the ones that have been successful in the context of SREU in the past (for example word equations). The methods that we use instead involve finite (tree) automata techniques, and the decidability proofs provide precise computational complexity bounds. The main results are 1) monadic SRR with ground rules is PSPACE-complete, and 2) balanced SRR with ground rules is EXPTIME-complete. The first result indicates the difference in computational power between fragments of SREU with ground rules and nonground rules, respectively, due to a straightforward encoding of word equations in monadic SREU (with nonground rules). The second result establishes the decidability and precise complexity of the largest known subfragment of nonmonadic SREU.