On-the-sphere compression of omnidirectional videos is a very promising approach. First, it saves computational complexity as it avoids to project the sphere onto a 2D map, as classically done. Second, and more importantly, it allows to achieve a better rate-distortion tradeoff, since neither the visual data nor its domain of definition are distorted. In this paper, the on-the-sphere compression [1] for omnidirectional still images is extended to videos. We first propose a complete review of existing spherical motion models. Then we propose a new one called tangent-linear+t. We finally propose a rate-distortion optimized algorithm to locally choose the best motion model for efficient motion estimation/compensation. For that purpose, we additionally propose a finer search pattern, called spherical-uniform, for the motion parameters, which leads to a more accurate block prediction. The novel algorithm leads to rate-distortion gains compared to methods based on a unique motion model.