Safety-Critical Path Planning with LQR-CBF-RRT* in 2D Dynamic Environments

Master's Thesis, M.Sc. in Applied Artificial Intelligence, IU University of Applied Sciences
Benjamin Müller · Supervisor: Dr. Usman Akhtar · December 2025

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What This Is

This thesis takes the LQR-CBF-RRT* motion planning framework (Yang et al., 2025), which only works with static obstacles, and extends it to handle dynamic environments with moving circular obstacles. The planner combines RRT* sampling, LQR steering, and a time-varying Control Barrier Function (CBF) that predicts obstacle positions and rejects unsafe edges during tree expansion and rewiring.

The evaluation answers three questions:

1. Safety: Does the planner keep collision rates below 1% over 3,000 roll-outs? Yes. Zero collisions observed; Wilson 95% upper bound at 0.13%.
2. Overhead: What does the CBF layer cost? Median planning time goes up ~8.6x while path length stays about the same.
3. Sensitivity: How do the CBF gain α, nominal speed v_nom, and extension length step_len affect results? 1,799 out of 1,800 successes across an 18-cell parameter grid. Clearance is mostly driven by speed and extension length, not CBF gain.

What I Built (vs. the Base Framework)

The starting point is the open-source LQR-CBF-RRT* implementation by Yang et al., which only handles static obstacles. I added:

• Time-varying CBF safety filter that predicts obstacle motion along candidate edges using constant-velocity kinematics
• Time-stamped tree expansion where nodes carry arrival times that propagate through rewiring
• Dynamic obstacle model with configurable count, speed, and radius ranges
• Experiment harness for batch execution of 3,000+ roll-outs with structured logging and CSV export
• Replay and audit tooling: JSON artifacts for collision cases, CLI-based replay, animation export, per-case diagnostics
• Paired evaluation protocol with seed-matched baseline comparison and Wilson confidence intervals

All of this is on the dyn-tvcbf branch of my fork.

Repo Structure

The code lives in my fork. This repo has the thesis document and links to the implementation:

Resource	Link
Thesis PDF	master_thesis_lqr_cbf_rrt.pdf
Implementation	Leg0shii/LQR_CBF_rrtStar (branch dyn-tvcbf)
Original framework	mingyucai/LQR_CBF_rrtStar
Base paper	Yang et al., 2023, arXiv:2304.00790

Key Files (on dyn-tvcbf branch)

├── LQR_CBF_rrtStar_linear.py   # Planner core: RRT* expansion, rewiring, tree structure
├── LQR_planning.py              # Discrete LQR steering with optional CBF safety layer
├── CBFsteer.py                  # CBF-QP with dynamic obstacle prediction
├── experiments.py               # Batch experiment driver (3000+ roll-outs)
├── replay_experiment.py         # Collision replay and animation tool
├── plotting.py                  # Visualization utilities
├── env.py                       # Environment and obstacle configuration
├── linear_dynamic_model/        # Linear system experiments
├── nonlinear_dynamic_model/     # Nonlinear system experiments
├── results/                     # Experiment logs and CSV summaries
└── prisma_tables/               # PRISMA literature review records

Results

Metric	Value
Collision rate (3,000 roll-outs, v_max = 1.0 m/s)	0.00% (Wilson 95% upper: 0.13%)
Median planning time overhead	~8.6x vs. LQR-RRT* baseline
Median path length difference	~1.7 m (not significant)
Sensitivity grid success rate	1,799 / 1,800 (99.94%)

Tech Stack

Python 3.10 · NumPy · SciPy · Gurobi (QP solver) · Matplotlib

Citation

@mastersthesis{mueller2025safety,
  title   = {Safety-Critical Path Planning with LQR-CBF-RRT* in 2D Dynamic Environments},
  author  = {M{\"u}ller, Benjamin},
  school  = {IU University of Applied Sciences},
  year    = {2025},
  month   = {December},
  type    = {Master's Thesis}
}

Extends the work of:

@inproceedings{yang2025lqrcbfrrt,
  title     = {LQR-CBF-RRT*: Safe and Optimal Motion Planning},
  author    = {Yang, Guang and Cai, Mingyu and Ahmad, Ahmad and Prorok, Amanda and Tron, Roberto and Belta, Calin},
  booktitle = {2025 American Control Conference (ACC)},
  pages     = {3700--3705},
  year      = {2025}
}