Precise control of movement across various robotic embodiments is a longstanding challenge. Modeling complex robotic structures requires significant time, dedicated sensors, and manual labeling. Recent advancements in 4D representation learning offer an interesting alternative: can we leverage 4D features lifted from 2D observations to densely map a robot's spatial movement based on the action signal? To this end, we propose Act4R, a framework that builds on 3D foundation model to predict a robot's 3D scene flow with any given action. Act4R introduces a Spatial-Action Encoder, which encodes the current 2D robot state and its action signals into embeddings that represent future robot state in 3D. Then, both the current and future spatial embeddings are decoded into point maps to estimate action-conditioned scene flow. To effectively learn the mapping between action and future state, Act4R introduces spatial-action fusion in its transformer-based encoder, and a spatial alignment loss that guides the encoder for future flow prediction. After training on some samples of a given embodiment, Act4R can perform inverse control by optimizing the signal to achieve the desired scene flow. We evaluate Act4R on a public real-world dataset and synthetic data collected in simulator. Experiments on various robotic platforms show that Act4R outperforms the previous SoTA in scene flow prediction. Our method also performs better in inverse control, both on the action accuracy metric and in closed-loop control in simulation.

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