Case Studies

This section describes the four main case studies that demonstrate RTD-RAX's capabilities compared to standard RTD.

Study 1: Gap Scenario

The canonical gap scenario places two rectangular obstacles symmetrically about y = 0 at x = 0.75 m, leaving a gap of 0.619 m. This tests whether the planner can find a trajectory through a narrow corridor.

FRS Version	Tracking Error	Reachable Set	Can Pass Gap
standard	Accounted for (conservative)	Larger	No
noerror	Removed	Smaller	Yes

The standard FRS is too conservative to navigate the gap. The noerror FRS finds a feasible path, and immrax verification confirms the planned trajectory is genuinely safe under the specified uncertainty.

Narrow gap scenario

Narrow gap scenario. Left: feasible parameter space (k) (safe: white, unsafe: pink) with selected trajectory (k^{*}) in green. Right: certifiably safe trajectory through the gap.

Gap Scenario Animation: Standard RTD Versus RTD-RAX

Animated Case 1 comparison. Left: standard RTD remains infeasible because the conservative FRS blocks the corridor. Right: RTD-RAX uses the noerror FRS plus immrax verification to certify a safe path through the gap.

Reproduce

cd docker/
make manuscript-case1-gap-suite

This regenerates the current saved Case 1 animation and figure family under figures/Study1_Gap/.

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Study 2: Angled Obstacles with Repair

This case study uses angled obstacles that require more 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.

The repair process:

1. Solve RTD with noerror FRS for candidate trajectory parameters k.
2. Verify with immrax under uncertainty/disturbance bounds.
3. If unsafe: speed-backoff (reduce k2) and/or CEGIS buffer tightening, then re-verify.
4. Execute only verified-safe candidates.

Angled Obstacle Comparison With Immrax Verification and Repair

Angled obstacle comparison under shared execution disturbance: Standard RTD succeeds conservatively, Non-Inflated RTD fails to make progress, and RTD-RAX reaches the goal with online Immrax verification and repair.

Angled Obstacle Repair View Under RTD-RAX Only

RTD-RAX repair event in the angled-obstacle scenario: an unsafe candidate tube is rejected, a repaired tube is selected, and execution continues safely to the goal. The gif is paused when a trajectory is rejected/repaired so the repair is evident visually in the animation. In reality it's almost instant 😎.

Reproduce

cd docker/
make rtd-case2-suite
make manuscript-case2-suite

The Case 2 animate targets save path-length JSON summaries automatically in case_study_outputs/ and in figures/Study2_AngledObs/.

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Study 3: Disturbance Compare

A randomized multi-gap course with disturbance patches compares standard RTD (executing directly under disturbance) against RTD-RAX (noerror FRS + immrax verification with measured disturbance bounds + repair).

Results

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

Standard RTD collides with an obstacle on cycle 3 because it cannot account for runtime disturbances. RTD-RAX detects the collision risk via immrax, repairs the trajectory, and reaches the goal safely.

Disturbance Comparison Animation

Disturbance comparison: standard RTD (top) hits the first obstacle block early in the rollout because the realized disturbance pushes it into the gate, while RTD-RAX (bottom) uses immrax verification and repair to avoid the block and finish the course safely.

Reproduce

cd docker/
make rtd-disturbance-compare
make manuscript-disturbance-gallery