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

2026-04-02

AI Summary

This work addresses the challenge of performing complex manipulation tasks in cluttered environments under stringent geometric and temporal constraints. Existing approaches struggle to support gradient-based optimization due to their reliance on non-differentiable geometric operations. To overcome this limitation, we propose the first end-to-end differentiable framework based on symbolic spatiotemporal logic. By constructing smooth, auto-differentiation-compatible tensorized geometric primitives—such as signed distances, intersections, containment, and directional relations—the method enables a differentiable mapping from high-level semantic specifications to low-level geometric configurations. Notably, it operates without external discrete solvers and facilitates both parallel trajectory optimization and direct backpropagation from demonstrations to learn logical parameters. Experiments demonstrate the approach’s effectiveness and scalability in optimizing trajectories under strict spatiotemporal constraints.

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.Code Available: https://github.com/plen1lune/DiffSpaTiaL

Problem

• Spatio-Temporal Logic
• Differentiable Programming
• Geometric Constraints
• Manipulation Tasks
• Gradient-based Optimization

Innovation

• Differentiable Spatio-Temporal Logic
• Geometric Temporal Logic
• Tensorized Geometric Primitives
• End-to-End Differentiability
• Symbolic Reasoning