Differentiable SpaTiaL: Symbolic Learning and Reasoning with Geometric Temporal Logic for Manipulation Tasks

TL;DR

Differentiable SpaTiaL introduces a fully tensorized, end-to-end differentiable toolbox for symbolic spatio-temporal logic, enabling gradient-based optimization and learning for manipulation tasks involving complex geometric and temporal constraints.

Contribution

It is the first to provide a fully differentiable, autograd-compatible geometric primitives and logic framework for manipulation tasks, bridging discrete spatial reasoning with continuous optimization.

Findings

Enables parallel trajectory optimization under spatio-temporal constraints.

Allows learning of spatial logic parameters directly from demonstrations.

Validates effectiveness and scalability through experiments.

Abstract

Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based optimization is limited by non-differentiable geometric operations. Existing differentiable temporal logics focus on the robot's internal state and neglect interactive object-environment relations, while spatial logic approaches that capture such interactions rely on discrete geometry engines that break the computational graph and preclude exact gradient propagation. To overcome this limitation, we propose Differentiable SpaTiaL, a fully tensorized toolbox that constructs smooth, autograd-compatible geometric primitives directly over polygonal sets. To the best of our knowledge, this is the first end-to-end differentiable symbolic spatio-temporal logic toolbox. By analytically deriving differentiable relaxations of key spatial predicates--including signed distance, intersection, containment, and directional relations--we enable an end-to-end differentiable mapping from high-level semantic specifications to low-level geometric configurations, without invoking external discrete solvers. This fully differentiable formulation unlocks two core capabilities: (i) massively parallel trajectory optimization under rigorous spatio-temporal constraints, and (ii) direct learning of spatial logic parameters from demonstrations via backpropagation. Experimental results validate the effectiveness and scalability of the proposed framework.