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The HR 8799 system uniquely harbors four young super-Jupiters whose orbitscan provide insights into the system's dynamical history and constrain themasses of the planets themselves. Using the Gemini Planet Imager (GPI), weobtained down to one milliarcsecond precision on the astrometry of theseplanets. We assessed four-planet orbit models with different levels ofconstraints and found that assuming the planets are near 1:2:4:8 periodcommensurabilities, or are coplanar, does not worsen the fit. We added theprior that the planets must have been stable for the age of the system (40 Myr)by running orbit configurations from our posteriors through $N$-bodysimulations and varying the masses of the planets. We found that only assumingthe planets are both coplanar and near 1:2:4:8 period commensurabilitiesproduces dynamically stable orbits in large quantities. Our posterior of stablecoplanar orbits tightly constrains the planets' orbits, and we discussimplications for the outermost planet b shaping the debris disk. A four-planetresonance lock is not necessary for stability up to now. However, planet pairsd and e, and c and d, are each likely locked in two-body resonances forstability if their component masses are above $6~M_{\rm{Jup}}$ and$7~M_{\rm{Jup}}$, respectively. Combining the dynamical and luminosityconstraints on the masses using hot-start evolutionary models and a system ageof $42 \pm 5$~Myr, we found the mass of planet b to be $5.8 \pm0.5~M_{\rm{Jup}}$, and the masses of planets c, d, and e to be$7.2_{-0.7}^{+0.6}~M_{\rm{Jup}}$ each.

Dynamical Constraints on the HR 8799 Planets with GPI