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
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 Rt 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 dsafe
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. ↑
Viol. ↓
Coll. ↓
Abort ↓
Jerk ↓
GoalDist ↓
MinDist ↑
Plan ms ↓
(%)
(%)
(%)
(%)
(m/s 3 )
(m)
(m)
μ/σ/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 dmin at t≈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.
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.
Taha Bouzid, Shuhao Qi, Mircea Lazar +1
Control Systems Group, Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
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.
Yiqi Zhao, Taekyung Kim, Hideki Okamoto +4
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
Motion planning for autonomous vehicles often requires satisfying multiple conditionally conflicting specifications. In situations where not all specifications can be met simultaneously, minimum-violation motion planning maintains system operation by minimizing violations of specifications in accordance with their priorities. Signal temporal logic (STL) provides a formal language for rigorously defining these specifications and enables the quantitative evaluation of their violations. However, a total ordering of specifications yields a lexicographic optimization problem, which is typically computationally expensive to solve using standard methods. We address this problem by transforming the multi-objective lexicographic optimization problem into a single-objective scalar optimization problem using non-uniform quantization and bit-shifting. Specifically, we extend a deterministic model predictive path integral (MPPI) solver to efficiently solve optimization problems without quadratic input cost. Additionally, a novel predicate-robustness measure that combines spatial and temporal violations is introduced. Our results show that the proposed method offers an interpretable and scalable solution for lexicographic STL minimum-violation motion planning within a single-objective solver framework.
Patrick Halder, Lothar Kiltz, Hannes Homburger +2
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