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

Licheng Luo; Kaier Liang; Cristian-Ioan Vasile; Mingyu Cai

cs.ROApr 3, 2026

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

Authors: Licheng Luo, Kaier Liang, Cristian-Ioan Vasile, Mingyu Cai

Organizations: UC Riverside · Lehigh University

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.

Figures & tables

Fig. 1 : Overview of Differentiable SpaTiaL . Our framework replaces discrete geometry engines with a fully tensorized architecture, enabling end-to-end trajectory optimization under formal specifications.

Framework / Toolbox	Temporal	Spatial	Differentiable	Batch-native tensorized
Diff-TL (e.g., STLCG [ 12 , 13 ] )	✓	$\times$	✓	✓
SpaTiaL [ 4 ]	✓	✓	$\times$	$\times$
Diff. Collision (e.g., DiffCol [ 21 ] )	$\times$	✓	✓	$\times$
Diff-SpaTiaL (Ours)	✓	✓	✓	✓

TABLE I : Feature comparison of our toolbox against existing formal logic and geometric frameworks.

Panel of Figure 3 — Fig. 2 : Smooth spatial and temporal operators form a differentiable robustness computation graph.

Fig. 3 : Differentiable penetration depth via Smooth SAT. Exact discrete overlap is relaxed into a smooth, differentiable repulsion field.

Panel of Figure 5 — Fig. 4 : Differentiable Signed Distance Field (SDF) via boundary sampling. Soft aggregation ensures informative gradients in both colliding and non-colliding states.

Fig. 5 : Geometric interpretation of compositional predicates (left: EnclIn, right: leftOf). Spatial relations are resolved as differentiable algebraic operations over vertex tensors.

Panel of Figure 7 — Fig. 6 : Qualitative spatio-temporal trajectory optimization using differentiable spatial semantics. From left to right: initial trajectory (violates), iteration 30, final trajectory (satisfies), and robustness analysis.

Panel of Figure 8 — Fig. 7 : Learning spatial specification parameters from demonstrations via robustness backpropagation. From left to right: oblique, xy, and xz views.

Start $\to$ Pick			Pick $\to$ Place
Obs	Formula	$\boldsymbol{\varepsilon}$	Obs	Formula	$\boldsymbol{\varepsilon}$
$O_{1}$	$F_{I_{1}}(\mathrm{RightOf})$	$4.25$	$O_{1}$	$G_{I_{2}}(\mathrm{RightOf})$	$3.31$
$O_{1}$	$F_{I_{1}}(\mathrm{Behind})$	$1.87$	$O_{1}$	$F_{I_{2}}(\mathrm{InFrontOf})$	$3.77$
$O_{2}$	$F_{I_{1}}(\mathrm{LeftOf})$	$5.25$	$O_{2}$	$F_{I_{2}}(\mathrm{InFrontOf})$	$4.73$
$O_{2}$	$F_{I_{1}}(\mathrm{Above})$	$2.55$	$O_{2}$	$F_{I_{2}}(\mathrm{Above})$	$2.52$
$O_{3}$	$F_{I_{1}}(\mathrm{LeftOf})$	$5.83$	$O_{3}$	$F_{I_{2}}(\mathrm{Behind})$	$3.66$

TABLE II : Discovered spatio-temporal specification with learned margins; phase 1 (

I_{1}=[0,59]

) and phase 2 (

I_{2}=[60,118]

). All predicates are of the form

(\texttt{arm},O_{k})

Fig. 8 : Geometric accuracy of differentiable predicates. We compare distance, signed distance, penetration, and enclosure robustness against Shapely across randomized convex polygons and transforms.

Fig. 9 : Effect of the smoothing temperature

\tau

and boundary sampling density on signed distance and its gradient.

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pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems

Bennett Dogbey, Hemanth Manjunatha

Dept. of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK USA

Signal Temporal Logic Safe Robot Navigation

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Autonomous robots operating in uncertain environments must satisfy complex temporal and safety specifications despite stochastic dynamics and sensing noise. While Signal Temporal Logic (STL) offers robustness measures for gradient-based optimization, existing extensions either lack differentiability or ignore belief-space uncertainty. We introduce pdSTL (probabilistic differentiable Signal Temporal Logic), a framework that unifies probabilistic semantics with differentiable robustness over belief trajectories. pdSTL employs interval-valued probabilistic semantics to compute conservative satisfaction bounds, propagated compositionally through the STL syntax tree. We formulate the temporal robustness evaluation as a recurrent, LSTM-style unfolding of STL operators, enabling linear-time, differentiable monitoring suitable for end-to-end trajectory optimization. We validate pdSTL on simulated obstacle avoidance, lane-change maneuvers, and real-world Crazyflie quadcopter flight experiments under aerodynamic disturbances. Results demonstrate that pdSTL achieves efficient optimization with formal probabilistic guarantees, significantly outperforming deterministic differentiable STL in maintaining safety margins under real-world uncertainty.

Abstract

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