SphLA: Spherical Latent Action Flow Matching for Humanoid Whole-Body Control

Behavioral foundation models (BFMs) provide pretrained whole-body policies for humanoid robots, supporting diverse behaviors through a shared policy. The latent representation of a Forward-Backward BFM conditions whole-body control. We use this representation as a generative action space, translating high-level task specifications into latent actions executed by the BFM's actor network without synthesizing kinematic motion references. However, these latent actions lie on a hypersphere, whose geometry conventional Euclidean flow matching does not preserve. To address this limitation, we introduce SphLA, spherical latent-action flow matching, which formulates conditional generation on this hypersphere. Our approach combines geodesic probability paths, tangent velocity prediction, and exponential-map integration to preserve the latent norm throughout sampling. SphLA supports both language and vision-language conditioning through independently trained action experts that share the same spherical body latent space and actor network. Experiments show that SphLA preserves spherical latent geometry and improves text–action alignment over Euclidean flow matching. Real-robot experiments validate language-directed humanoid whole-body control.