Efficient Hybrid Hard--Soft Signal Temporal Logic MPC for Driving under Multimodal Uncertainty

Kumar Manas; Adrian Paschke

cs.ROOct 11, 2026

Efficient Hybrid Hard--Soft Signal Temporal Logic MPC for Driving under Multimodal Uncertainty

Authors: Kumar Manas, Adrian Paschke

Organizations: Technical University Darmstadt, Germany · Freie Universität Berlin, Germany · Freie Universität Berlin and Fraunhofer FOKUS, Berlin, Germany

Abstract

Motion planning for automated vehicles in interactive traffic has to simultaneously ensure safety, respect traffic rules, and maintain comfortable progress under multi-modal behavior uncertainty. Imposing all these requirements as hard constraints reduces feasibility, whereas treating them all as costs allows safety-critical requirements to be traded against progress. We propose a hybrid hard--soft Signal Temporal Logic (STL) MPC framework that optimizes a single shared ego trajectory against every retained behavior hypothesis: minimum-distance and speed predicates are imposed as explicit hard constraints, while soft STL robustness terms shape clearance, comfort, and interaction quality within the resulting feasible driving corridor. This retains robustness to discrete interaction modes without branch-specific contingent ego policies. We further identify substantial symbolic graph-construction overhead and introduce reachability-certified relevance screening and vectorized symbolic construction to reduce computational cost while preserving the feasible set and objective. Closed-loop Monte Carlo evaluation over six highway and intersection scenarios shows the highest strict success rate among the compared planners with no observed collisions or rule violations, at roughly a quarter of the planning latency of the unoptimized formulation. An ablation study separates the two layers: the hard constraints define the safety envelope, while the soft terms improve clearance and motion quality within it.

Figures & tables

Method	Multi-modal	Ego policy	STL	Hard/soft split	Postpone / recovery	Computational mechanism
Robust scenario MPC [ 5 ]	✓	mode-dependent	$-$	$-$	$-$	tube–scenario formulation
Contingency MPC [ 6 ]	partial	parallel	$-$	$-$	contingency	parallel prediction horizons
Branch MPC [ 1 ]	✓	tree	$-$	$-$	implicit	scenario-tree structure
Learning-based BMPC [ 13 ]	✓	tree	$-$	$-$	✓	online scenario selection
Risk-aware BMPC [ 7 ]	✓	tree	$-$	$-$	$-$	iterative min–max risk update
Game-theoretic BMPC [ 14 ]	✓	tree	$-$	$-$	$-$	tailored tree solver

TABLE I : Comparison with representative uncertainty-aware and STL-MPC methods. “Multi-modal” indicates discrete behavior hypotheses; “hard/soft split” distinguishes non-relaxable constraints from preferences.

Fig. 1 : Proposed Hybrid STL-MPC pipeline. Multi-modal predictions constrain a single shared ego trajectory; no scenario tree or branch-specific controls are built, and mode probabilities enter only the postponement trigger. Minimum distance and speed limit are explicit constraints; soft costs shape preferred clearance, speed, and scenario-specific intersection-yield and no-passing behavior. Screening is per (participant, timestep): terms beyond the ego reachable bound

R_{t}

are provably inactive and removed before symbolic construction, leaving the feasible set and objective unchanged (Proposition 1 ). Postponement is evaluated before each solve and bypasses it when a shield certifies non-negative clearance; solver failure triggers a fallback that drops soft costs but retains every hard constraint.

Rule	Hard role	Soft role
Distance	Explicit minimum clearance against all retained hypotheses	Preferred clearance $d_{\mathrm{safe}}$
Speed	Explicit upper-speed constraint	Preferred upper speed threshold
Intersection yield	Covered by generic hard distance safety	Gated crossing-traffic clearance cost
No passing	–	Zone-gated soft cost plus a separate high-weight lateral-deviation penalty

TABLE II : STL terms entering the reported benchmark. “Hard” denotes an explicit NLP constraint; soft terms enter the quadratic objective. Distance thresholds are adapted to scenario geometry as described in Sec. V .

Method	Succ. $\uparrow$	Viol. $\downarrow$	Coll. $\downarrow$	Abort $\downarrow$	Jerk $\downarrow$	GoalDist $\downarrow$	MinDist $\uparrow$	Plan ms $\downarrow$
	(%)	(%)	(%)	(%)	(m/s 3 )	(m)	(m)	$\mu/\sigma/p95$
Single-mode MPC	39.3	18.7	15.3	34.3	4.162	1.270	4.266	122.9/156.1/491.6
Risk-Aware BMPC [ 7 ]	45.0	9.3	2.7	44.3	3.763	1.402	4.125	409.7/487.4/648.0
Game-Theoretic BMPC [ 14 ]	49.0	1.3	0.0	49.7	5.541	1.392	4.073	318.6/ 115.7 /538.0
Soft-only	93.3	6.7	3.0	0.0	4.090	1.400	5.603	122.7/237.4/464.4
Hard-only	98.7	0.0	0.0	0.0	4.366	2.048	4.393	99.4 / 114.0 / 347.4

TABLE III : Closed-loop results pooled over six scenarios. Outcome rates are computed over all 300 episodes per method. Jerk, final goal distance, and minimum distance are averaged over completed episodes (non-aborted) only. Planning time reports mean/std./95th percentile over all MPC calls. Lower is better for jerk, goal distance, and time; higher minimum distance indicates larger executed clearance. Best result is in bold , followed by underlined as second best.

Fig. 2 : Measured effects of both contributions. (a) Minimum executed clearance in a challenging left-turn episode. Soft-only uses the same soft STL costs as Hybrid but no explicit NLP constraints; it crosses

d_{\min}

t\approx 1.15

s and collides, whereas Hybrid and Hard-only remain safe. Across 300 episodes, Soft-only has 6.7% violations and 3.0% collisions, versus 0.0% for Hybrid (Table III ). Hybrid and Hard-only both maintain the hard clearance; aggregate clearance differences appear in Table III . (b) Fraction of hard-distance terms retained by exact relevance screening, ranging from 17% on the highway merge to 72% at the dense intersection. By Proposition 1 , removed terms are inactive, preserving the feasible set and objective.

Scenario	Hybrid	Soft	Hard	Single-mode	Risk-BMPC	Game-BMPC
Intersection	100/0	98/2	100/0	0/0	0/0	6/0
Left turn	100/0	62/38	100/0	96/4	80/2	90/8
Highway merge	100/0	100/0	100/0	40/60	30/54	98/0
Multi-modal int.	100/0	100/0	100/0	0/48	94/0	0/0
No passing	96/0	100/0	92/0	0/0	0/0	0/0
Stress highway	100/0	100/0	100/0	100/0	66/0	100/0

TABLE IV : Per-scenario strict success / recorded violation rate (%). Each entry is “success / violation” over 50 trials.

Configuration	Build	Solve	Total
	(ms)	(ms)	(ms)
Baseline	234.9	484.6	733.7
+ relevance screening	181.3	339.0	526.3
+ vectorized construction	22.5	93.0	118.7

TABLE V : Staged latency ablation on the challenging left-turn scenario, paired over identical seeded episodes. Each row adds one component; the represented feasible set and objective are unchanged throughout.

STL Planning

See all 17 papers

Autonomous Driving with Priority-Ordered STL Specifications Under Multimodal Uncertainty
Jun 18, 2026Taha Bouzid, Shuhao Qi, Mircea Lazar +1STL Motion Planning Motion Planning
Control Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
Safety-aware Model Predictive Path Integral Control with Signal Temporal Logic
Aug 25, 2026Yiqi Zhao, Taekyung Kim, Hideki Okamoto +4STL Motion Planning Safe Motion Planning
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 +1
Lexicographic Minimum-Violation Motion Planning using Signal Temporal Logic
Apr 22, 2026Patrick Halder, Lothar Kiltz, Hannes Homburger +2STL Motion Planning Motion Planning
School of Computation, Information and Technology, Technical University of Munich, 85748 Garching, Germany · 2ZF Friedrichshafen AG, 88046 Friedrichshafen, Germany · Institute of System Dynamics, HTWG Konstanz, 78562 Konstanz, Germany

Jun 18, 2026J/K move · Enter open · S save

Autonomous Driving with Priority-Ordered STL Specifications Under Multimodal Uncertainty

Taha Bouzid, Shuhao Qi, Mircea Lazar, Sofie Haesaert

Control Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

STL Motion Planning Motion Planning Autonomous Driving Planning

Open paper AskPDF arXiv

Autonomous vehicles must plan trajectories that satisfy a multitude of requirements on safety, passenger comfort, and compliance with traffic rules. However, in safety-critical scenarios, it is not always possible to satisfy all requirements simultaneously, necessitating their prioritization based on importance. At the same time, in these safety-critical scenarios, the uncertainty in trajectory predictions of the surrounding traffic, such as other vehicles and pedestrians, should be explicitly accounted for. In this work, we propose an uncertainty-aware trajectory planning framework that incorporates a predefined lexicographic ordering over Signal Temporal Logic (STL) specifications that stays valid under uncertainty. We implement this formulation with Model Predictive Path Integral (MPPI) control and we demonstrate the effectiveness of our method on simulation scenarios, showing that our framework efficiently handles conflicting objectives under realistic multi-modal uncertainty.

Abstract

STL Planning

Explore similar work

Autonomous Driving with Priority-Ordered STL Specifications Under Multimodal Uncertainty

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

Lexicographic Minimum-Violation Motion Planning using Signal Temporal Logic