CoGeL | Constraint-to-Geometry Learning

Abstract

Robots are often deployed in environments where safety must be learned from experience rather than specified in advance. Demonstrations, operator interventions, near misses, and failed rollouts can reveal which behaviors are risky, but they provide only local evidence. The key challenge is not only to identify risky behavior, but to make that knowledge shape the distance used for long-horizon planning. Existing goal-conditioned methods can compose short-horizon behaviors into longer plans, but they typically assume that the relevant cost structure is already known.

We introduce CoGeL, a constraint-to-geometry learning method that learns a soft safety signal from robot data and incorporates it directly into the goal-reaching distance used for planning. This makes the learned distance safety-aware rather than treating safety as a separate classifier, hard mask, or post-hoc filter. We show that errors in the learned safety signal induce bounded changes in the navigation geometry, that route consistency supports composition through intermediate waypoints, and that, under a calibration condition, the learned safety scores bound the expected number of safety violations. Experiments across wheeled and legged navigation tasks show that CoGeL reduces unsafe events while maintaining strong long-horizon goal-reaching performance.

Constraint-to-Geometry Learning

CoGeL overview. Robot evidence is used to learn a local safety signal, which reshapes the directed goal-reaching distance used for long-horizon waypoint planning.

How CoGeL Works

CoGeL turns local demonstrations, failures, and interventions into a safety-aware planning geometry.

1

Robot Evidence

Collect demonstrations, exploratory rollouts, failures, interventions, and safety-monitor events.

2

Safety Signal

Learn a soft risk estimate \[ \hat{\chi}_{\psi}(s,a)\in[0,1]. \]

3

Directed Geometry

Incorporate the learned safety signal into the Bellman cost to learn a directed distance.

4

Long-Horizon Planning

Use the learned distance as a waypoint cost so local behaviors can form longer safe routes.

Safety-aware cost

\[ \tilde c_{\psi}(s,a,s') = c(s,a,s') + \lambda\hat{\chi}_{\psi}(s,a). \]

Risky actions become more expensive before route composition. The penalty changes smoothly with the learned safety estimate.

Route consistency

\[ d_{\theta}(s,g) \lesssim d_{\theta}(s,w) + d_{\theta}(w,g). \]

This encourages the learned distance to compose through intermediate waypoints over longer routes.

Navigation Scenarios

Simulation. Hidden terrain hazards and directional risks are not provided to the policy as an explicit safety map.

CoGeL real-world robot navigation scenarios

Real-world navigation. CoGeL favors routes around loose debris, slippery terrain, and pavement hazards.

Results

0.91 Wheeled navigation success

0.72 Wheeled violations / episode

0.76 Wheeled LH-8× success

0.68 Legged LH-8× success

Long-horizon generalization and safety-efficiency tradeoff. CoGeL maintains stronger success as test horizons increase, while the safety weight controls the tradeoff between success and violations.

Main Navigation Results

Method	SR ↑	CV ↓	SPL ↑	LH-4× ↑	LH-8× ↑
GCBC	0.79	4.35	0.61	0.34	0.16
Lag-PPO	0.80	1.85	0.58	0.45	0.24
CRL	0.85	3.05	0.67	0.52	0.31
QRL	0.89	2.75	0.73	0.65	0.43
Safety+Plan	0.83	1.55	0.64	0.61	0.39
CoGeL	0.91	0.72	0.76	0.84	0.76

Wheeled navigation results. SR: success rate; CV: constraint violations per episode; SPL: success weighted by path length.

Theory and Empirical Checks

Safety → Geometry

If \[ \|\hat{\chi}_{\psi}-\chi^\star\|_{\infty} \le \epsilon_{\chi}, \] then \[ \|d^\star_{\hat{\chi}} - d^\star_{\chi^\star}\|_{\infty} \le \frac{\lambda}{1-\gamma} \epsilon_{\chi}. \]

Waypoint Composition

For \(w_0=s,\ldots,w_N=g\), \[ d_{\theta}(s,g) \le \sum_{i=0}^{N-1} d_{\theta}(w_i,w_{i+1}) + (N-1)\epsilon_{\triangle}. \]

Directional Hazards

If the directional cost gap is \(\Delta>0\), every symmetric metric incurs an approximation error of at least \[ \frac{\Delta}{2} \] in one direction.

Empirical checks of CoGeL theoretical results

Empirical checks of the theory. Safety-estimation error yields bounded geometry error, route error increases with composed waypoint segments, and calibrated risk scores bound expected safety violations.

Why Directed Geometry?

Many robot hazards are directional. Entering slippery terrain, climbing onto unstable debris, or approaching a narrow region may cost more than leaving it.

CoGeL therefore learns a directed distance:

\[ d^\star(s,g) \neq d^\star(g,s). \]

A symmetric metric must assign the same value to both directions and therefore cannot represent this difference.

Entering the hazard can cost more than leaving it.

Safety-Aware Waypoint Planning

The learned directed distance makes a safer waypoint route preferable even when the direct route is shorter.

Real-Robot Evaluation

Physical navigation trials with a Boston Dynamics Spot and Clearpath Jackal.

Robot	Method	Trials	SR ↑	Safety events / trial ↓
Spot	Safety+Plan	20	0.75	0.45
Spot	CoGeL	20	0.85	0.20
Jackal	Safety+Plan	20	0.80	0.40
Jackal	CoGeL	20	0.90	0.15

Challenging Cases

Slippery stair approach. CoGeL favors an entry route that avoids the local slippery region.

Confined stepping failure. Planning geometry alone cannot replace accurate local foothold and contact control.

Limitations

CoGeL depends on the quality and coverage of the evidence used to learn the safety signal. Sparse, biased, or poorly timed labels can cause \(\hat{\chi}_{\psi}(s,a)\) to miss hazards or over-penalize safe behavior. CoGeL does not guarantee zero violations, and the safety estimate can become poorly calibrated under distribution shift. Long-horizon performance also depends on waypoint coverage; a global route can fail if the graph misses a narrow safe corridor even when the local distance is accurate.

BibTeX

@inproceedings{hossain2026cogel,
  title     = {{CoGeL}: Constraint-to-Geometry Learning from Demonstrations and Failures for Safe Long-Horizon Robot Navigation},
  author    = {Jumman Hossain and Nirmalya Roy},
  booktitle = {10th Conference on Robot Learning (CoRL)},
  year      = {2026},
  address   = {Austin, TX, USA}
}

CoGeL: Constraint-to-Geometry Learning from Demonstrations and Failures for Safe Long-Horizon Robot Navigation