01Motivation
A CBF-QP safety filter keeps a robot inside the constraint set \(\mathcal C = \{x : c(x)\ge 0\}\) by modifying a nominal control input:
\[ \begin{aligned}
u^{*} = \operatorname*{arg\,min}_{u\,\in\,\mathcal U}\ \ &
\tfrac12\lVert u - u_{\mathrm{nom}}\rVert^2 \\[-1pt]
\text{s.t.}\ \ & L_f h(x) + L_g h(x)\,u + \alpha\bigl(h(x)\bigr) \ge 0
\end{aligned} \]
- Under input limits, valid CBFs are hard to design and often overly conservative.
- Grid-based HJ reachability is exact but scales exponentially with the state dimension.
- PDE-only neural CBFs tend to converge to trivial solutions.
02The maximal safe set
For dynamics \(\dot x = \phi(x,u) = f(x) + g(x)u\) with a convex input polytope \(\mathcal U\), the HJ safety value function
\[ V(x) = \sup_{u(\cdot)\in\,\mathcal U}\ \inf_{t\ \ge\ 0}\ c\bigl(x(t)\bigr) \]
defines the maximal safe set \(\mathcal S = \{x : V(x) \ge 0\}\), the largest controlled invariant subset of \(\mathcal C\).
We learn a neural CBF \(V_\Theta\) whose safe set approximates \(\mathcal S\).03Neural CBF from the HJB equation
\(V_\Theta\) is trained on the stationary HJB variational inequality. The Hamiltonian is linear in \(u\), so its maximum is attained at a vertex of \(\mathcal U\):
\[ \min\Bigl\{\, c(x) - V(x),\ \ \max_{u\,\in\,\mathcal V}
\nabla V(x)^\top\!\phi(x,u) \,\Bigr\} = 0 \]
04Model architecture and training
\(V_\Theta\) is parametrized as a non-negative residual subtracted from the constraint:
\[ V_\Theta(x) = c(x) - r_\Theta(x), \qquad r_\Theta(x) \ge 0 \]
The learned safe set always lies inside
\(\mathcal C\).
Training objective:
\[ \mathcal L(\Theta) = \lambda\,\mathcal L_{\mathrm{PDE}}(\Theta)
+ (1-\lambda)\,\mathcal L_{\mathrm{data}}(\Theta) \]
- \(\mathcal L_{\mathrm{PDE}}\): HJB residual on unlabelled states
- \(\mathcal L_{\mathrm{data}}\): regression on vertex-search labels
05Vertex-restricted control search
Labels come from a finite-horizon search over sequences of vertex controls:
\[ V^{K,B}_{\Delta t}(x) = \max_{u_0,\dots,u_{K-1}\,\in\,\mathcal V}\ \
\min_{0\,\le\, k\,\le\, K}\ c(x_k) \]
- Beam search keeps the best \(B\) partial trajectories at each of the \(K\) steps.
- GPU-parallel over states, beams and vertices; no PDE solver is needed.
06Evaluation metrics
Predictions are checked with closed-loop rollouts:
- \(\rho_{\mathrm{FS}}\) false-safe: predicted safe, but the rollout leaves \(\mathcal C\)
- \(\rho_{\mathrm{FU}}\) false-unsafe: predicted unsafe, but the rollout stays safe
- \(\eta_{\mathrm{eff}}\) effective safe volume: ratio of states predicted and validated safe
07Better supervision data
08Recovering the maximal safe set
09Results on 15 systems (\(\eta_{\mathrm{eff}} = \tfrac{\mathrm{vol}(\hat{\mathcal S})} {\mathrm{vol}(\mathcal X)}(1-\rho_{\mathrm{FS}})\))
PDE-only
Full-control data
VertexCBF (ours)
Inverted Pendulum (2,1)
1D Double Integrator (2,1)
2D Vertical Drone (2,1)
Dubins Car (3,1)
2D Double Integrator (4,2)
Kinematic Bicycle (4,2)
Cart-Pole (4,1)
Dynamic Unicycle (5,2)
Relative Unicycle (5,2)
3D Double Integrator (6,3)
3-DoF Manipulator (6,3)
Landing Rocket (7,2)
Quadruped Trunk (9,4)
6-DoF AUV (12,6)
Quadrotor (13,4)
00.51.0
Mean of five seeds; \((n_x, n_u)\) in brackets.
15/15
systems with the highest or tied-highest effective safe volume
≤ 1.2 %
false-safe rate on every system; baselines exceed 20 % or return an
empty safe set on six
96.4 %
IoU with the true safe set on the inverted pendulum, vs. 33.5 % and
1.5 % for the baselines
10Hardware experiments
- A mobile robot follows an MPC planner, and the learned CBF filters its controls in real time using live pedestrian detections.
- One or two pedestrians cross the robot's path in each trial; all 10 trials were collision-free.
11Conclusion
- Searching only over the vertices of \(\mathcal U\) provides effective supervision without a PDE solver.
- Limitations: no formal guarantees yet; the search cost grows with \(|\mathcal V|\) and \(K\).
- Future work: error guarantees, systems with many control vertices, and uncertain dynamics.