Fast GPU evaluation of differentiable Signed Distance Fields for robotics via tensor decompositions

Signed Distance Fields (SDF) are widely used in robotics for collision checking, reactive control and trajectory optimization, but high-fidelity batched evaluation on GPUs remains challenging. This work proposes a low-rank formulation based on tensor decompositions of Bernstein polynomial SDF models. In particular, link-wise coefficient tensors are represented in low-rank tensor formats, namely canonical polyadic (CP) and tensor train (TT), yielding compact and differentiable implicit models with analytical evaluation of both the distance field and its gradient. This is especially relevant for robotics applications in which collision handling often requires not only distance queries but also gradient information to generate repulsive actions, deformation directions and optimization updates. A roofline-inspired analysis is introduced to interpret the runtime behavior of dense and compressed models across polynomial orders and batch sizes. Experiments on SDFs representing the different links of a robot manipulator as implicit shapes show that low-rank significantly reduces runtime and memory with respect to the dense model, while preserving sufficient geometric accuracy for collision-aware applications. In particular, CP provides the best nominal runtime in most tested settings, whereas TT offers a more favorable memory scalability trade-off. Reactive collision-avoidance examples further confirm that the proposed models provide gradients accurate enough for real-time motion generation.