This is the final version of the article. Available from the American Astronomical Society / IOP Publishing via the DOI in this record. ; We present an analysis of the orbital motion of the four substellar objects orbiting HR 8799. Our study relies on the published astrometric history of this system augmented with an epoch obtained with the Project 1640 coronagraph with an integral field spectrograph (IFS) installed at the Palomar Hale telescope. We first focus on the intricacies associated with astrometric estimation using the combination of an extreme adaptive optics system (PALM-3000), a coronagraph, and an IFS. We introduce two new algorithms. The first one retrieves the stellar focal plane position when the star is occulted by a coronagraphic stop. The second one yields precise astrometric and spectrophotometric estimates of faint point sources even when they are initially buried in the speckle noise. The second part of our paper is devoted to studying orbital motion in this system. In order to complement the orbital architectures discussed in the literature, we determine an ensemble of likely Keplerian orbits for HR 8799bcde, using a Bayesian analysis with maximally vague priors regarding the overall configuration of the system. Although the astrometric history is currently too scarce to formally rule out coplanarity, HR 8799d appears to be misaligned with respect to the most likely planes of HR 8799bce orbits. This misalignment is sufficient to question the strictly coplanar assumption made by various authors when identifying a Laplace resonance as a potential architecture. Finally, we establish a high likelihood that HR 8799de have dynamical masses below 13 MJup, using a loose dynamical survival argument based on geometric close encounters. We illustrate how future dynamical analyses will further constrain dynamical masses in the entire system.