RTD-RAX

Official Documentation Website for this project is found here.

The arXiv preprint of the RTD-RAX paper may be found here.

RTD-RAX is a runtime-assurance extension of Reachability-based Trajectory Design (RTD) that replaces conservative offline reachable sets with fast online safety certification via mixed-monotone reachability (immrax).

Left: Narrow gap scenario. Feasible parameter space k (safe: white, unsafe: red) with selected trajectory k^* in green, and certifiably safe trajectory through the gap. Right: Online repair process under disturbance, showing successive trajectory corrections.

RTD is a provably safe, real-time motion planning framework that precomputes Forward Reachable Sets (FRS) offline and uses them online to optimize for collision-free trajectories. The catch: because high-fidelity models are too expensive for reachable-set computation, RTD uses simplified models and inflates the FRS with worst-case tracking-error bounds. This makes the planner overly conservative — it rejects safe trajectories, triggers unnecessary braking, and cannot handle disturbances (wind, ice, slippage) that weren't anticipated offline.

RTD-RAX fixes this by splitting the problem: RTD's offline reachable sets handle fast candidate generation without conservative inflation, while a separate online verifier built on mixed-monotone reachability certifies each candidate under the actual measured uncertainty and disturbance bounds. If a candidate can't be certified safe, a repair procedure modifies it until a safe alternative is found.

1. Plan — use the uninflated FRS to rapidly generate candidate trajectories.
2. Verify — certify each candidate online via mixed-monotone reachability under current conditions.
3. Repair — if unsafe, modify the candidate and re-verify before execution.

Key Results

• Safe trajectory planning under a priori unknown disturbances
• Reduction in conservatism compared to standard RTD, enabling navigation through narrow corridors and around angled obstacles
• Elimination of needless fail-safe maneuvers present in standard RTD
• Real-time verification and repair under measured disturbance bounds
• Demonstrated on a 4D unicycle model with three case studies

Quick Start

All case studies run inside a Docker container with pre-installed dependencies (JAX, immrax, matplotlib, etc.).

Option A: Pre-built Image (Fastest)

git clone https://github.com/evannsm/rtd-rax.git && cd rtd-rax/docker
make pull             # pulls evannsmc/rtd-rax:latest from DockerHub
make run_gui          # starts container with X11 forwarding for plots

Option B: Build from Source

git clone https://github.com/evannsm/rtd-rax.git && cd rtd-rax/docker
make build
make run_gui          # starts container with X11 forwarding for plots

Run Examples

make rtd-gap          # standard FRS — too conservative, no path found
make rtd-gap FRS=noerror   # noerror FRS — feasible path found

For GUI plotting on Linux/X11, allow Docker access first:

xhost +local:docker

After finishing, revoke access:

xhost -local:docker

Case Studies

All make targets are run from the docker/ directory.

Study 1: Gap Scenario

Two rectangular obstacles leave a narrow gap. Standard RTD is too conservative to pass; RTD-RAX uses the noerror FRS plus immrax verification to certify a safe path through.

make manuscript-case1-gap-suite

Study 2: Angled Obstacles with Repair

Angled obstacles require complex trajectories. The noerror FRS proposes candidates that may be unsafe under uncertainty — immrax catches the collision risk, and the hybrid repair loop finds safer alternatives.

make rtd-case2-suite
make manuscript-case2-suite

Study 3: Disturbance Compare

A randomized multi-gap course with disturbance patches. Standard RTD collides on cycle 3; RTD-RAX detects the risk via immrax, repairs, and reaches the goal safely.

Planner	Outcome	Cycles	Repairs	Mean / p95 Compute
Standard RTD	Collision (cycle 3)	3	--	10.5 ms / 21.9 ms
RTD-RAX	Goal reached	19	3	10.5 ms / 37.4 ms

make rtd-disturbance-compare
make manuscript-disturbance-gallery

Repository Structure

rtd_rax/
├── turtlebot_rtd_numpy/          # Core RTD-RAX implementation (NumPy/SciPy)
│   ├── one_shot_rtd.py           # Basic single-shot planner
│   ├── one_shot_rtd_gap.py       # Gap scenario with immrax verification
│   ├── rtd_gap_journey.py        # Multi-step receding-horizon replanning
│   ├── rtd_gap_journey_compare.py        # Standard vs noerror comparison with repair
│   ├── rtd_angled_obstacle_compare.py    # Angled obstacle scenario with repair
│   ├── rtd_random_disturbance_compare.py # Disturbance course comparison
│   ├── immrax_verify.py          # immrax reachability verification
│   ├── disturbance_case_study_utils.py   # Shared utilities for case studies
│   ├── frs_loader.py             # FRS .mat file loader
│   ├── constraints.py            # FRS polynomial constraint builder
│   ├── cost.py                   # Trajectory cost function
│   ├── polynomial_utils.py       # Polynomial evaluation utilities
│   ├── geometry_utils.py         # Obstacle geometry helpers
│   ├── trajectory.py             # Trajectory generation
│   └── turtlebot_agent.py        # Turtlebot agent simulation
├── python_preprocessed_frs/      # Precomputed FRS data files (.mat)
├── docker/                       # Docker setup and make targets
│   ├── Dockerfile
│   ├── makefile                  # All experiment and figure targets
│   ├── requirements.txt
│   └── README.md                 # Detailed Docker reference
└── README.md

Make Targets

Core Experiments

Target	Description
rtd-gap	Gap scenario (FRS=standard or FRS=noerror)
rtd-gap-verify	Gap scenario with immrax verification
rtd-journey-gap-compare	Journey comparison with verification and repair
rtd-case2-suite	Case 2 representative + two-repair outputs
rtd-disturbance-compare	Disturbance course comparison

Manuscript Figures

Target	Description
manuscript-case1-gap-suite	Case 1 GIF/PDF family
manuscript-case2-suite	Case 2 GIF/PDF family
manuscript-disturbance-gallery	Disturbance comparison outputs
manuscript-figures	Regenerate all manuscript assets

Common Parameters

Variable	Default	Description
FRS	standard	FRS variant (standard or noerror)
UNCERTAINTY	0.01	immrax positional uncertainty (m)
DISTURBANCE	0.0	Bounded additive disturbance
JOURNEY_VERIFY	0	Enable immrax verification (1 to enable)
JOURNEY_REPAIR	0	Enable hybrid repair (1 to enable)

# Example: gap verification with custom uncertainty
make rtd-gap-verify FRS=noerror UNCERTAINTY=0.05 DISTURBANCE=0.01

See docker/README.md for the full target and parameter reference.

Container Lifecycle

Command	Description
make pull	Pull pre-built image from DockerHub
make build	Build the Docker image from source
make run	Start container (headless)
make run_gui	Start container with X11 forwarding
make start	Restart a stopped container
make stop	Stop the running container
make attach	Attach a shell to the running container

Dependencies

Installed automatically by Docker:

• NumPy, SciPy, Matplotlib, Shapely
• JAX (CPU)
• immrax — interval-arithmetic reachability library

License

MIT

Website

This project is part of the evannsmc open-source portfolio.