The sign
Will the policy stay safe?
For the ideal value, Vπ(x) ≤ 0 means the policy keeps the safety signal nonpositive for all future time. A positive value means a future violation.
Accepted to CoRL 2026
λ-ReachabilityGeometric-Horizon Safety Bellman Equations for Humanoid Safety
Learning to anticipate a humanoid’s worst future safety condition—before a fall or collision occurs.
On the real robotUnitree G1
See the safety prediction in the full video
01 / The idea in motion
Look ahead.
Recognize risk earlier.
A robot can look stable now and still be on a trajectory toward failure. λ-Reachability learns a safety value for a fixed policy that estimates the worst safety signal along its future trajectory.
02 / Safety value functions
Safety beyond
the current state.
The instantaneous signal ℓ tells us whether a constraint is violated now. The safety value Vπ looks along the future trajectory under the robot’s policy, capturing both whether a violation will occur and the safety margin.
The safety value function
Vπ(xt) = supk ≥ t ℓ(xk)
For a fixed policy π, take the largest safety signal over its entire future trajectory. λ-Reachability learns an estimate of this value.
ℓ(x): what is happening nowInstantaneous balance or collision constraint.
Vπ(x): what lies aheadWorst safety signal along the future trajectory.
The margin
How close is the boundary?
The value locates the worst future condition relative to the zero safety threshold. A negative value closer to zero leaves less margin in the chosen safety signal.
The warning
See risk before it happens.
The learned value can turn positive while the current signal is still negative, anticipating a fall or collision before the constraint is violated.
03 / The main idea
Learn from the future.
One segment at a time.
λ-Reachability learns the worst future safety signal from randomly sized trajectory segments. A geometric horizon connects short, bootstrapped updates to longer max-over-trajectory targets, so each update can bring future safety information back to the current state.
- 01
Sample a horizon
Draw n from a geometric distribution. λ controls how far the update looks into the future.
- 02
Observe the worst signal
Take the maximum safety signal over the sampled trajectory segment.
- 03
Bootstrap probabilistically
Include the terminal value with probability δⁿ; otherwise use a lower-bound terminal value.
Explore the horizon
One parameter.
A different look ahead.
Increase λ to shift probability toward longer trajectory segments.
0 · one step0.99 · longer horizons
steps on averageE[n] = 1 / (1 − λ)
Probability per five-step group
n = 1–5Rollout horizon96–100
Beyond 100 stepsCombined probability · P(n ≥ 101) = λ¹⁰⁰
Bars show equal five-step groups; the remaining probability is reported separately above. This is the untruncated distribution. Training uses finite, truncated horizons.
A connection to undiscounted reachability. The ideal operator is a contraction for δ < 1, and its fixed point approaches the undiscounted safety value as λ → 1. These theoretical properties do not by themselves certify a learned neural model on hardware. See §3 and Appendix A
04 / Hardware demonstrations
Watch the robot.
Read its safety value.
Play a trial to watch the robot and its safety value evolve together. The animated traces show the learned monitor anticipating risk as a push or an incoming ball changes the robot’s situation.
Look for the warning before the violation. A positive learned value predicts future risk. When the blue curve crosses zero before the red curve, it anticipates a violation while the current constraint is still satisfied.
- Learned safety value Vπ
- Current safety signal ℓ
- Obstacle distance · collision trials
Balance / G1 FlatUnsafe outcome
A warning before the fall
The learned value crosses zero before the instantaneous balance signal. Watch how the warning relates to the loss of balance.
Balance / G1 FlatSafe outcome
Recovery with a safety margin
Both recorded traces stay below zero as the robot recovers from the push.
Avoidance / G1 CollisionUnsafe outcome
A warning before contact
The learned value becomes positive while the ball approaches. The instantaneous signal crosses zero when contact occurs.
Avoidance / G1 CollisionSafe outcome
A warning without a collision
The robot avoids contact and the instantaneous signal stays negative. Positive value predictions illustrate conservative warnings in these trials.
Use the video controls to play, pause, seek, or open both views in fullscreen. The curves replay with the camera footage at the labeled slow-motion speed. Collision timing offsets are estimated from the recorded events. Blue shading marks intervals when the learned value is positive while the current signal is nonpositive; red shading marks an observed violation. A safe outcome can still include positive predictions from the learned monitor.
BibTeX
@inproceedings{chen2026lambdareachability,
title = {$\lambda$-Reachability: Geometric-Horizon Safety Bellman Equations for Humanoid Safety},
author = {Rui Chen and Shangtao Li and Yifan Sun and Changliu Liu},
year = {2026},
booktitle = {Conference on Robot Learning},
url = {https://arxiv.org/abs/2606.16022},
website = {https://intelligent-control-lab.github.io/lambda_reach/},
youtube = {https://www.youtube.com/watch?v=sW9JjJ73NJ4},
code = {https://github.com/intelligent-control-lab/lambda_reach}
}Conference on Robot Learning, 2026.