**4. Conclusions**

This experiment implemented and compared the 12 DCM to Euler Angle rotations using a variable step size. The effects on accuracy, step size, and timing were observed, and the simulation results showed that the DCMs were classified into symmetric and non-symmetric rotations. The non-symmetric rotations were easier to correlate and compare, while the symmetric rotations were not, limiting analysis. Furthermore, no one rotation was the ideal in the analyzed categories. This is beneficial, because trade space analysis can be conducted to determine accuracy, timing, and other high priority design criteria to select the appropriate DCM. The lowest *roll* mean error is obtained by using any of the 123, 132, 213, 231, 312, or 321 rotation sequences, while the lowest *pitch* mean error *cannot be achieved by the ubiquitous 321 sequence*, *instead the 132 sequence must be used*; while the lowest *yaw* mean error may be achieved with the 213, 231, and 321 sequences. Standard deviations show similar options for selecting different rotation sequences for specific applications. Regarding computational efficiency, the 232 sequence was best, followed by the 313, and then the 121 sequence. The ubiquitously accepted standard 321 sequence was found to be fifth fastest, with four other rotation sequences bearing less computational burden. Novel illustrations include the fact that one of the ubiquitous sequences (the "313 sequence") has degraded relative accuracy measured by mean and standard deviations of errors, but may be calculated faster than the other ubiquitous sequence (the "321 sequence"), while a lesser known "231 sequence" has comparable accuracy and calculation-time. Evaluation of the 231 sequence also illustrates the originality of the research. These novelties are applied to spacecraft attitude control in this manuscript, but can equally be applied to robotics, aircraft, and surface and subsurface vehicles.

Lastly, future research would refine the correlation for symmetric rotations, but furthermore experimental validation will be performed on free-floating spacecraft simulator hardware at the Naval Postgraduate School. The validation will be performed by duplicating one of the specifically cited technical applications (e.g., any of the technical applications in [35–45]) seeking to validate performance improvement.

**Author Contributions:** B.S. and T.S. conceived and designed the research; B.S. and A.R. conceptualized and developed the methodology, performed the experiments, reviewed the data and validated the results; T.S. performed literature review, wrote the final manuscript, and managed the peer review process. Authorship has been limited to those who have contributed substantially to the work reported.

**Funding:** This research received no external funding. No sources of funding apply to the study. Grants were not received in support of our research work. No funds were received for covering the costs to publish in open access, instead this was an invited manuscript to the special issue.

**Conflicts of Interest:** The authors declare no conflict of interest.
