cs.ROAug 25, 2026

Safety-aware Model Predictive Path Integral Control with Signal Temporal Logic

Authors: Yiqi Zhao, Taekyung Kim, Hideki Okamoto, Bardh Hoxha, Jyotirmoy V. Deshmukh, Lars Lindemann, Georgios Fainekos

Organizations: Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA 90089, USA · Department of Robotics, University of Michigan, Ann Arbor, MI 48109, USA · Toyota Motor North America R&D, Ann Arbor, MI 48105, USA · Automatic Control Laboratory, ETH Zürich, Zürich, Switzerland

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Abstract

Safety-aware motion planning remains a challenge in robotics, especially when missions are time-critical and are under complex specifications. In this paper, we propose safety-aware-stl-mppi, a computationally efficient sampling-based receding-horizon planning framework designed to promote satisfaction of constraints expressed in Signal Temporal Logic (STL). Our approach encodes discrete-time STL formulas into candidate time-varying control barrier functions (CBF), which are integrated into a model predictive path integral (MPPI) controller. Our method inherits the benefits of low computational cost from an efficiently parallelizable sampling based planner and utilizes CBF for constraints expressed in STL. We compare against several MPPI baselines using four artificial Mars Rover planning case studies with a diverse environment and cost setups, where we show our method consistently achieving high safety and efficiency. We show a quadcopter planning experiment with NVIDIA Isaac Lab.

Figures & tables

Fig. 1 : Safety-aware-stl-mppi: The top row highlights the STL motion planning problem with an STL formula ϕ\phi . Our solution is on the second row, where we rely on a time-varying CBF hϕh^{\phi} , which encodes the region consistent with the STL constraint with its time-varying zero-super level set (shown in orange). We plan with MPPI using CBF informed weighted rollout, and project any weighted rollout (dashed green) violating the CBF into the safety region (solid green). The planning works in receding-horizon with a closed-loop trajectory (in yellow) satisfying ϕ\phi . The drawings in the figure are hypothetical. We show drawings in a case study in Section V .
FormulaEncoding Rule (which we follow when constructing the CBF)
πμ\pi^{\mu}∀x∈D,hπμ(x,τ0)≥0\forall x\in D,h^{\pi^{\mu}}(x,\tau_{0})\geq 0 implies (x,τ0)⊨πμ(x,\tau_{0})\models\pi^{\mu} .
G[a,b]πμG_{[a,b]}\pi^{\mu}∀t′∈σ(a,b),∀x∈D,hG[a,b]πμ(x,t′)≥0 implies μ(x)≥0\forall t^{\prime}\in\sigma(a,b),\forall x\in D,h^{G_{[a,b]}\pi^{\mu}}(x,t^{\prime})\geq 0\text{ implies }\mu(x)\geq 0 .
F[a,b]πμF_{[a,b]}\pi^{\mu}∃t′∈σ(a,b),∀x∈D,hF[a,b]πμ(x,t′)≥0 implies μ(x)≥0\exists t^{\prime}\in\sigma(a,b),\forall x\in D,h^{F_{[a,b]}\pi^{\mu}}(x,t^{\prime})\geq 0\text{ implies }\mu(x)\geq 0 .
μ1U[a,b]μ2\mu_{1}U_{[a,b]}\mu_{2}hμ1U[a,b]μ2(x,t):−min~(h1(x,t),h2(x,t))h^{\mu_{1}U_{[a,b]}\mu_{2}}(x,t)\coloneq\widetilde{\text{min}}(h_{1}(x,t),h_{2}(x,t)) where ∃t′∈σ(a,b),∀x∈D,h2(x,t′)≥0 implies μ2(x)≥0\exists t^{\prime}\in\sigma(a,b),\forall x\in D,h_{2}(x,t^{\prime})\geq 0\text{ implies }\mu_{2}(x)\geq 0
 and ∀t′′∈σ(0,t′),∀x∈D,h1(x,t′′)≥0 implies μ1(x)≥0\forall t^{\prime\prime}\in\sigma(0,t^{\prime}),\forall x\in D,h_{1}(x,t^{\prime\prime})\geq 0\text{ implies }\mu_{1}(x)\geq 0 .
G[a,b]μ1∧μ2G_{[a,b]}\mu_{1}\wedge\mu_{2}hG[a,b]μ1∧μ2(x,t):−min~(h1(x,t),h2(x,t))h^{G_{[a,b]}\mu_{1}\wedge\mu_{2}}(x,t)\coloneq\widetilde{\text{min}}(h_{1}(x,t),h_{2}(x,t)) where ∀t′∈σ(a,b),∀x∈D,h1(x,t′)≥0 implies μ1(x)≥0\forall t^{\prime}\in\sigma(a,b),\forall x\in D,h_{1}(x,t^{\prime})\geq 0\text{ implies }\mu_{1}(x)\geq 0
TABLE I : Candidate CBF functions for STL encoding
Fig. 2 : Example Trajectories Planned with safety-aware-stl-mppi (Limited Actuation).
TABLE II : Section V-A Evaluation Results.
Fig. 3 : Quadcopter Simulation Example

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