Learning Reachability-Based Safety Representations for Real-Time Robot Navigation in Unknown and Dynamic Environments

Bojan Derajić

Figure 1: Mobile robot navigation in an unknown warehouse environment shared with other agents and perceived through local onboard sensing. The space outside the perception field is unknown, and the robot must react to new observations while maintaining safety margins under its dynamics and actuation limits.

Figure 2: Unsafe states for a mobile robot avoiding a static obstacle and a dynamic obstacle crossing its path. The shaded regions are still collision-free, but neither full braking nor maximal steering to either side prevents collision. For the static obstacle, the region grows roughly with the braking distance. For the moving obstacle, it is additionally shaped by the relative motion. A controller that monitors only current distance (dashed trajectories) may enter this region, but a dynamics-aware controller (solid trajectory) evades while recovery remains possible.

Figure 3: Typical mobile robot navigation architecture. Perception converts sensor data into maps, state estimates, and obstacle information. A global planner provides a route toward the task goal, a local planner converts this route and the latest observations into executable local motion, and a feedback controller tracks the resulting reference. An optional safety filter (dashed) supervises the nominal command and modifies it when it violates the specified safety condition.

Figure 4: Qualitative positioning of the reviewed method families and this dissertation. The horizontal axis indicates adaptation to new local observations, and the vertical axis indicates the extent to which safety reasoning accounts for robot dynamics and actuation limits. Placements represent typical formulations, and the learning-based safety filters include both learned barrier functions and learned reachability values. This dissertation studies reachability-informed safety constraints across static and dynamic environments and across MPC and safety-filter interfaces.

Figure 5: Organization of the dissertation. After the introduction and the preliminaries, Parts I and II group the technical chapters by controller interface, and within each part the development progresses from static to dynamic environments. Part III varies a third design choice, the origin of the reachability supervision. The concluding chapter compares the methods of the three parts.

Figure 1: Braking example in the state space of the double integrator

\dot{p}=v

,

\dot{v}=u

with

\lvert u\rvert\leq\bar{u}

and a wall at

p<0

. The failure set

\mathcal{F}

(dark) collects the collision states, and the constraint set

\mathcal{C}

is its complement. The horizontal axis is the position

p

and the vertical axis is the velocity

v

. Within

\mathcal{C}

, the states to the left of the braking parabola

p=v^{2}/(2\bar{u})

(shaded red) are collision-free at the current instant but cannot avoid future collision, because their distance to the wall is smaller than the distance needed to brake. The maximal safe set

\mathcal{S}^{\star}

(light) excludes them. The dashed curves show two trajectories under maximal braking that start with the same approach speed, and both are horizontal translates of the braking parabola. The left one starts inside

\mathcal{C}\setminus\mathcal{S}^{\star}

and reaches the wall with nonzero speed (cross), while the right one starts in

\mathcal{S}^{\star}

and stops at a positive distance from the wall (open circle).

Figure 2: Braking example, continued. Constraint function

c

and stationary value function

V_{\infty}

along a slice of the state space with constant approach speed

v_{0}<0

. The value function runs below the constraint function by exactly the braking distance

v_{0}^{2}/(2\bar{u})

, which is the residual

r=c-V_{\infty}

of Equation 44 . Its zero crossing is shifted by the braking distance in front of the wall, and the shaded region between the two zero crossings contains the states that satisfy the constraint but are dynamically unable to avoid collision.

Figure 3: Offline–online pipeline shared by the methods of this dissertation. Offline, a solver computes HJ value functions

V_{w}

for a set of training conditioning inputs, and these solutions supervise the learned safety model. Chapters 2 , 3 , 4 and 5 obtain them from a grid-based solver, while Chapter 6 replaces this block by vertex-restricted search combined with a physics-informed loss. Online, the current conditioning input

w

selects the scalar field

\hat{V}(\,\cdot\mid w)

whose value and gradient the optimization-based controller evaluates. The compact main network takes the robot state

x

as input and is conditioned on

w

, either through parameters generated by the optional hypernetwork (dashed) once per update, as in Chapters 2 , 3 and 4 , or directly when the hypernetwork is omitted. The main network is evaluated and differentiated at every query.

Figure 1: Effect of the terminal constraint on an MPC local planner that approaches a static obstacle, shown at two consecutive planning steps. Each red-shaded region is a slice of the backward reachable tube (BRT) of the observed obstacle, that is, of the set

\mathcal{X}\setminus\mathcal{S}^{\star}(o_{k})

of states from which collision can no longer be avoided under the actuation limits. The slice depends on the heading, so a solid outline belongs to the current state and a dashed outline to the terminal state. In the lower row the dotted body marks the position at step

k

. Left: the terminal constraint is distance based, so every state predicted at step

k

keeps a nonnegative distance to the obstacle and the planner continues straight ahead. At step

k+1

the robot has entered the BRT, and every admissible input sequence collides at some later instant. Right: the reachability-based terminal constraint

x_{N\mid k}\in\mathcal{S}^{\star}(o_{k})

makes the planner steer away already at step

k

, so the robot remains outside the BRT one step later.

Figure 2: Architecture of the NTC-MPC local planner. From the current sensor data, an SDF is computed and used both as the geometric stage constraint of the MPC problem and as the conditioning input

w_{k}

of the hypernetwork. The hypernetwork, a convolutional backbone with a fully connected head, outputs the parameters of the compact main network, which represents the estimated value function

\hat{V}_{\Psi}(\,\cdot\mid w_{k})

. The nonlinear program (NLP) solver queries the main network for the value and gradient of the terminal constraint at every optimizer iteration, while the hypernetwork is evaluated only once per planning cycle.

Figure 3: Training procedure for the NTC model. Each SDF conditioning input

w

from the dataset is processed by the hypernetwork into the main-network parameters

\Theta(w)

, and all states of a state-space grid are propagated through the main network to predict the value function over the grid. HJ reachability analysis, using the same SDF as the constraint function

c(\,\cdot\mid w)

, provides the numerical target values

V_{\infty}(\,\cdot\mid w)

. The loss

\mathcal{L}

compares predicted and target value functions, and its gradient updates the hypernetwork parameters

\Psi

. The main-network parameters are outputs of the hypernetwork rather than free variables. The figure shows a single training pair, so the batch and grid indices of Equation 7 are omitted.

Observation (SDF) size	$100\times 100$ cells, $6\times 6$ $\mathrm{m}$
Dataset size (train/val)	$20\,000$ SDFs (80%/20%)
Hypernetwork	CNN backbone with fully connected head
Hypernetwork parameters	$\sim 4.8\times 10^{6}$
Main network hidden layers	$[32,32,32,16,16,16,8,8,8]$ , sine + SELU
Main network output	1 (value function)
Loss	RWMSE, $\alpha=1000$ , $\beta=10$

Table 1: NTC-MPC configuration shared by both robot models.

	Dubins car	Dynamic unicycle
State-space grid	$100\times 100\times 20$	$50\times 50\times 7\times 7\times 7$
Main network inputs	3	5
Main network parameters	$3601$	$3665$
Epochs / batch size	150 / 32	100 / 16
Training time (RTX 3090)	$\sim$ $7.5\text{\,}\mathrm{h}$	$\sim$ $18\text{\,}\mathrm{h}$

Table 2: NTC-MPC configuration specific to each robot model.

Figure 4: Learned safe set for a sample observation from the validation dataset (Dubins car). (a) Local occupancy grid

o

. (b) The corresponding SDF

c(\,\cdot\mid o)

, whose zero level set (black) outlines the obstacles. (c) Slice of the learned value function

\hat{V}_{\Psi}(\,\cdot\mid w)

at heading

\theta=\pi/2

, that is, for a robot at the marked position (dart) moving upward. Red shading marks the states predicted to be unsafe. The predicted safe-set boundary

\hat{V}_{\Psi}=0

(green) closely follows the numerical HJ boundary

V_{\infty}=0

(dashed). The unsafe region extends beyond the obstacle outlines, most visibly below the obstacles, which the robot approaches head-on at this heading.

Figure 5: Effect of the training objective for the Dubins car model, evaluated on the validation set. (a) Safe-set IoU over training epochs for models trained with RWMSE and MSE under identical settings and initialization. (b), (c) Confusion matrices of the safe/unsafe classification for RWMSE and MSE, with entries in percent of all grid states. RWMSE yields a consistently higher IoU and fewer misclassifications on both sides of the boundary.

Figure 6: Gazebo simulation environments. Left: warehouse environment, from which all training observations are collected. Right: house environment, used exclusively for evaluation to assess generalization under a distribution shift in obstacle geometry.

Figure 7: Success rates in the Monte Carlo simulation study for the two robot models (columns) and the two environments (rows), with 100 runs per configuration. The models are trained on warehouse data only and evaluated in both environments. The performance of the distance-based baselines degrades considerably as the prediction horizon becomes shorter, while the NTC-MPC method maintains higher success rates across all horizon lengths. The RWMSE-trained model also consistently matches or outperforms its MSE-trained counterpart.

Figure 8: NLP solve times of the three planners for (a) the Dubins car and (b) the dynamic unicycle as functions of the horizon length, pooled over both environments. Each box spans the interquartile range of the per-run mean solve times of the successful runs, the line marks the median, and the whiskers extend to 1.5 times the interquartile range. The neural terminal constraint adds an approximately horizon-independent offset over SDF-MPC, while the stage-wise DCBF-MPC constraints scale faster with the horizon. All values remain considerably below the

100\text{\,}\mathrm{ms}

planning period.

Figure 9: The ten static obstacle configurations used in the hardware experiments, viewed from the external camera. The robot starts below the lower edge and must reach a waypoint near the upper edge of the scenario image.

	Success rate [%]				Mean computation time [ $\mathrm{ms}$ ]
$N$ [steps]	5	10	20	30	5	10	20	30
SDF-MPC	0	0	30	70	2.65	4.55	6.26	6.54
DCBF-MPC	0	10	70	80	4.06	5.61	5.91	7.76
NTC-MPC (ours)	80	90	90	90	4.93	5.26	6.54	6.31

Table 3: Hardware experiment results with the Dubins car model over the ten obstacle configurations of Figure 9 .

Figure 10: Representative hardware experiment with the Dubins car model and horizon

N=10

(

1\text{\,}\mathrm{s}

). The upper row shows the overlaid robot motion for the three planners, where both baselines collide with the obstacle (red) while NTC-MPC avoids it (green). The plots show the time evolution of the predicted terminal value

\hat{V}_{\Psi}(x_{N\mid k}\mid w_{k})

and of the observed obstacle distance

c(x_{k}\mid o_{k})

, with the zero level marked by the dotted line. In this configuration, the obstacle distance of the baselines remains positive until shortly before the collision, while NTC-MPC begins to avoid the obstacle when its terminal value approaches zero, several seconds earlier.

Figure 1: Effect of the terminal constraint on an MPC local planner among moving obstacles, shown at two consecutive planning steps. The robot tracks the reference path to the right at constant speed and with a limited turning rate, while an obstacle crosses that path from the front left. Each red region collects the states from which a collision can no longer be avoided under the actuation limits. It extends along the obstacle velocity relative to the robot , so it is not aligned with the drawn velocity arrow, and the region at the terminal instant is the region at the current instant displaced and rotated according to the obstacle motion over the horizon. In the lower row the dotted outlines mark the positions at step

k

. Left: constraints formed from the current observation refer only to the unsafe region at the current instant,

\mathcal{X}\setminus\mathcal{S}_{0\mid k}

, so the planner continues straight ahead at step

k

. At step

k+1

its state lies inside that region, where continued straight motion and both extreme steering maneuvers end in the predicted obstacle. Right: the terminal constraint

x_{N\mid k}\in\mathcal{S}_{N\mid k}

of RNTC-MPC refers to the predicted geometry and is already active at step

k

, so the planner turns away and its plan remains feasible one step later.

Figure 2: RNTC-MPC architecture. Tracked obstacle positions and velocities are extrapolated over the MPC horizon and converted into a predicted SDF sequence. The stage-matched SDFs define the collision-avoidance constraints, while selected snapshots near the terminal stage form the conditioning input of the hypernetwork. The hypernetwork runs once per planning cycle and generates the main-network parameters. Inside the nonlinear program, the terminal constraint is the predicted terminal SDF minus the nonnegative main-network output, as in Equation 6 . The solver evaluates this assembled function and its derivatives repeatedly.

Figure 3: RNTC training procedure. Each obstacle trajectory is split at a reference instant. Selected snapshots before that instant form the conditioning input

w

of Equation 5 , which the hypernetwork converts into the main-network parameters

\Theta(w)

. The generated main network produces a nonnegative residual on the state-space grid. The subsequent trajectory segment defines the time-varying failure set used to compute the numerical HJ target

V(\,\cdot\mid w)

at the reference instant. Subtracting the predicted residual from the reference-instant SDF yields the value-function estimate used in the training loss. The figure shows a single training pair, so the batch and grid indices of Equation 10 are omitted. Signal names match Figure 3 .

Figure 4: Pointwise optimum

\hat{y}^{*}

of the CME loss Equation 12 as a function of the target

y

for different weights

\lambda

. For

\lambda=1

the loss reduces to MSE and the optimum is the identity (dashed). As

\lambda

decreases, predictions near the zero level set are displaced away from zero, while the sign, the zero crossing, and the behavior for large

\lvert y\rvert

are preserved.

Robot model	kinematic unicycle Equation 14
Sensing window	$8\times 8$ $\mathrm{m}$ , $100\times 100$ cells
Obstacles per scene	up to 4, speeds $\leq$ 1\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1}$$ , constant velocity
Condition (SDF snapshots)	$L=2$ , lags $(q_{1},q_{2})=(4,0)$ , $0.4\text{\,}\mathrm{s}$ apart
Dataset size	$40\,000$ SDF-sequence/value-function pairs
State-space grid	$100\times 100\times 30$ over $(p_{x},p_{y},\theta)$
Hypernetwork	CNN backbone with fully connected head

Table 1: RNTC-MPC model and training configuration.

Figure 5: Residual parameterization versus direct value regression at equal model capacity, evaluated on the validation set for three main-network sizes (

2790

,

4519

, and

6670

parameters). (a) Safe-set intersection over union (IoU) over training epochs, with the residual models in green and the direct-regression models in orange, and the line pattern indicating the network size. The residual models start higher, converge roughly five times faster, and end higher. (b) Final IoU after 100 epochs. The residual models reach a higher IoU at every size, and the smallest residual model outperforms the largest direct-regression model.

Figure 6: Qualitative comparison for two validation samples (a) and (b), each shown as a heading slice over the full sensing window and, connected by the dotted lines, as a magnified detail of the marked rectangle. Gray disks mark the predicted failure set, in which the terminal constraint function

c_{N\mid k}

is negative. The curves are the zero level sets of the numerical HJ target (dashed), of the direct-regression estimate (orange), and of the residual estimate (green). The direct-regression boundary enters the predicted failure set in several locations, while the residual estimate excludes that region by construction.

Figure 7: Effect of the training objective on the safe-set IoU, evaluated on the validation set over training epochs for the same residual model trained with the CME loss (

\lambda=0.1

) and with plain MSE under identical settings. CME converges faster, reaching the final MSE level after about 20 epochs, and ends at a higher IoU (

\approx 0.975

versus

\approx 0.967

).

Figure 8: Gazebo simulation environment. Six cylindrical obstacles with randomly initialized positions and velocity vectors (arrows) move at constant speed within a corridor and reverse direction at its edges. The robot must navigate from the start position to the goal position ahead of it, using only local observations.

Figure 9: Monte Carlo simulation results over 100 randomized scenarios. (a) Success rate against the prediction-horizon length. RNTC-MPC achieves the highest rate at every tested horizon and consistently exceeds direct-regression NTC-MPC. (b) Per-run mean NLP solve times over the scenarios completed by all planners, which remain considerably below the

50\text{\,}\mathrm{ms}

control-update period. Boxes span the interquartile range, lines mark the median, and whiskers extend to 1.5 times the interquartile range.

Figure 10: Motion efficiency in simulation over the scenarios completed by all planners. (a) Travel times from start to goal, which are similar across methods. (b) Mean travel time against success rate, with one marker per method and horizon. Higher success and shorter travel time are preferable, so the lower right region is favorable. RNTC-MPC achieves the highest success rate at every tested horizon with a mean travel time close to the lowest.

Figure 11: (a) Per-run mean and (b) per-run maximum lateral deviation from the reference path in the simulation study, shown over the horizon length for the scenarios completed by all planners. VO-MPC deviates most at the three shorter horizons, which reflects its evasive velocity constraints, while RNTC-MPC remains comparable to the geometric baselines. The deviations of all methods are larger at long horizons than at short ones, since a longer look-ahead triggers earlier and wider avoidance.

Figure 12: Executed trajectories over all 100 simulation runs for each planner at

N=10

, corresponding to a

1\text{\,}\mathrm{s}

prediction horizon, plotted as lateral deviation

d

against the distance along the reference path. Green denotes successful runs and red denotes collisions. SDF-MPC remains close to the reference but collides frequently, VO-MPC deviates widely from the reference, and the planners with learned terminal constraints achieve moderate deviations with fewer failures.

Method	Success rate [%]	$d_{\mathrm{mean}}$ [ $\mathrm{m}$ ]	$d_{\mathrm{max}}$ [ $\mathrm{m}$ ]	Opt. time [ $\mathrm{ms}$ ]	Travel time [ $\mathrm{s}$ ]
SDF-MPC	40.0	0.03 $\pm$ 0.01	0.08 $\pm$ 0.01	2.93 $\pm$ 0.25	13.03 $\pm$ 1.73
DCBF-MPC	70.0	0.11 $\pm$ 0.06	0.36 $\pm$ 0.22	3.09 $\pm$ 0.38	13.98 $\pm$ 0.93
VO-MPC	70.0	0.19 $\pm$ 0.13	0.53 $\pm$ 0.33	6.94 $\pm$ 2.61	14.67 $\pm$ 1.55
RNTC-MPC (ours)	100.0	0.04 $\pm$ 0.02	0.10 $\pm$ 0.08	4.20 $\pm$ 0.56	12.76 $\pm$ 0.89

Table 2: Hardware experiment results in the pedestrian-avoidance task, with 10 trials per method and horizon

N=10

, corresponding to

1\text{\,}\mathrm{s}

. Values are reported as mean

\pm

standard deviation where applicable.

Figure 13: Successful collision avoidance by RNTC-MPC during the hardware experiments with a prediction horizon of

1\text{\,}\mathrm{s}

. The robot slows down before the crossing pedestrian and resumes its path once the predicted pedestrian trajectory no longer constrains the terminal state.

Figure 14: Executed trajectories of the four planners in one representative hardware scenario with a single crossing pedestrian, with one run per planner under the same experimental setting, plotted as lateral deviation

d

against the distance along the reference path, with the dots along each trajectory colored by time. SDF-MPC fails to react and collides after about

2.5\text{\,}\mathrm{m}

(red cross). VO-MPC evades with a detour of up to

0.8\text{\,}\mathrm{m}

and DCBF-MPC with up to

0.16\text{\,}\mathrm{m}

, while RNTC-MPC resolves the encounter with a deviation of only a few centimeters by adjusting its timing along the path.

Figure 15: Onboard view during a hardware trial. Top left: boundary of the predicted terminal safe set relative to the predicted pedestrian motion. Bottom left: front-camera view. Right: detected pedestrian and predicted trajectory used to construct the SDF sequence.

Figure 1: Effect of the barrier candidate on a CBF-QP safety filter at three instants separated by a short interval

\delta

. The constant-speed Dubins car of Section 1 cannot brake and is driven toward a goal behind an obstacle. The red region is the slice at the current heading of the set

\mathcal{X}\setminus\mathcal{S}^{\star}(o)

from which a collision can no longer be avoided, and it changes only because the robot turns. Left: the barrier is the observed obstacle distance

c(\,\cdot\mid o)

, enforced through the second-order exponential CBF of Section 4 . The robot turns away only once the distance to the obstacle has become small. By then the state has crossed

\partial\mathcal{S}^{\star}(o)

, and at

t+2\delta

both extremal turns end in a collision. Right: the zero level set of

\hat{h}_{\Psi}(\,\cdot\mid w)\approx V_{\infty}(\,\cdot\mid o)

coincides with

\partial\mathcal{S}^{\star}(o)

, so the filter acts already at

t

and keeps the state inside

\mathcal{S}^{\star}(o)

, where an admissible input always remains.

Figure 2: The critical moment of an observation update. Geometry that the current observation does not contain is drawn faded and with dashed boundaries. (a) At

t_{j}

, a previously unobserved obstacle lies beyond the boundary of the sensing window of half-width

\rho

. Neither the failure set

\mathcal{F}

(grey) nor its BRT

\mathcal{B}_{\infty}(\mathcal{F})

(red) is part of the observation

o_{j}

. (b) Between two updates the robot travels at most

v_{\max}\Delta t_{o}

, and the window is recentered at the new robot position. The new observation thus gains a band of that depth beyond its previous boundary (dashed) and now contains most of the obstacle. Under the boundary-appearance assumption, previously unseen obstacles can appear only inside this band, and hence at least

\rho-v_{\max}\Delta t_{o}

away from the robot. Continued safety requires the state to lie outside

\mathcal{B}_{\infty}(\mathcal{F})

at the instant at which the barrier is replaced.

Figure 3: Architecture of the ORN-CBF safety filter. The discretized SDF

w_{j}

computed from the observation

o_{j}

is used twice, as the conditioning input of the hypernetwork, which generates the parameters

\Theta

of the main network representing the nonnegative residual

\hat{r}_{\Psi}

, and as the input of the interpolant representing the observed constraint function

c(\,\cdot\mid o_{j})

. Barrier value and gradient are assembled by subtraction as in Equation 9 , and the Lie derivatives

L_{f_{0}}\hat{h}_{\Psi}

and

L_{g}\hat{h}_{\Psi}

parameterize the CBF-QP Equation 13 together with

\hat{h}_{\Psi}

. The softplus output of the main network is not drawn as a separate stage. The hypernetwork runs once per observation update, and the main network and the interpolant are queried at control rate.

Figure 4: Closed-loop arrangement of the ORN-CBF safety filter for a robot that operates on local observations. A nominal controller of arbitrary design issues the command

u_{\mathrm{nom}}

. From the current observation

o_{j}

, ORN-CBF generates the barrier candidate

\hat{h}_{\Psi}(\,\cdot\mid w_{j})

once per observation update, as detailed in Figure 3 . The CBF-QP Equation 13 then evaluates this barrier at the measured state

x

at control rate, modifies the command minimally if needed, and forwards the filtered control

u^{\star}

to the robot. The dashed arrows indicate that the nominal controller may use the same two signals, on which the filter places no requirement.

	Dubins car	Planar double integrator
Training environment	warehouse	forest-like
Observations collected	$2500$	$2000$
After augmentation ( $\times 8$ )	$20\,000$	$16\,000$
Sensing window	$6\times 6$ $\mathrm{m}$ , $0.06\text{\,}\mathrm{m}$ res.	$6\times 6$ $\mathrm{m}$ , $0.075\text{\,}\mathrm{m}$ res.
HJ state-space grid	$100\times 100\times 30$	$80\times 80\times 20\times 20$
HJ computation (RTX 3090)	$\sim$ $1.17\text{\,}\mathrm{h}$	$\sim$ $16.3\text{\,}\mathrm{h}$

Table 1: ORN-CBF data generation and Hamilton–Jacobi targets for the two robot models.

Hypernetwork	CNN backbone with fully connected head
Main network hidden layers	$[32,32,32,16,16,16,8,8,8]$ , sinusoidal
Main network output	softplus (nonnegative residual)
Loss	RWMSE ( Section 3 )
Optimizer / epochs / batch	Adam / 100 / 16
Learning rate	$10^{-4}$ , reduced $10\times$ at epochs 85 and 95

Table 2: ORN-CBF configuration shared by both robot models.

	Dubins car	Planar double integrator
Main network inputs	3	4
Main network parameters	$3601$	$3633$
Training time (RTX 3090)	$\sim$ $0.5\text{\,}\mathrm{h}$	$\sim$ $2.9\text{\,}\mathrm{h}$

Table 3: ORN-CBF configuration specific to each robot model.

Figure 5: A learned ORN-CBF for the Dubins car on a validation observation. (a) The observed SDF

c(\,\cdot\mid o)

with the obstacle outlines

c=0

(black) and the robot position at heading

\theta=\pi/2

(dart, moving upward). (b) The numerical HJ target

V_{\infty}(\,\cdot\mid o)

at the same heading with its zero level set (dashed). Red shading marks the unsafe states, which extend beyond the obstacle outlines in the directions in which the robot approaches the obstacles head-on, and the orange square marks the central

16\times 16

positional patch on which the residual network is trained. (c) The learned barrier

\hat{h}_{\Psi}(\,\cdot\mid w)

on this patch whose zero level set (green) closely follows the HJ boundary. The asymmetry of the unsafe region reflects the turning limitation of the constant-speed vehicle.

Figure 6: Ground-robot simulation study. Left: scene from the Gazebo warehouse environment. Right: the corresponding local observation from which the ORN-CBF barrier is generated.

Figure 7: Success rates of the ground-robot simulation study over the MPC prediction-horizon length, with 100 randomized runs per configuration. The safety filters ORN-CBF and ON-CBF are combined with the plain SDF-MPC planner. Both outperform every MPC-based planner for

N\leq 20

, including the reachability-informed NTC-MPC of Chapter 2 , which reaches a similar rate at

N=30

. ORN-CBF achieves the highest rates overall.

Figure 8: Quadrotor simulation study. Left: scene from the Gazebo forest-like environment. Right: the corresponding local observation.

Figure 9: Outcomes of the quadrotor simulation study over 200 randomized scenarios per configuration, split into successes, timeouts, and collisions. (a) In the training domain, the learned filters succeed in more than

90\%

of the runs, compared with

58.5\%

for the hand-tuned ECBF. (b) In the out-of-domain environment with unseen obstacle radii, the success rates of the learned filters decrease by at most two percentage points, while that of the ECBF decreases to

45.0\%

.

Figure 10: Representative quadrotor scenario. (a) Executed motion under the LQR nominal controller with the ORN-CBF safety filter among the ten cylindrical obstacles (gray), colored by time. The filter steers the quadrotor around the obstacles with which the goal-directed LQR would otherwise collide. (b) Evolution of the barrier value

\hat{h}_{\Psi}(x(t)\mid w_{j})

, which decreases toward zero during the avoidance maneuvers and remains positive throughout the run.

Figure 11: Ground-robot hardware experiments. Left: the differential-drive robot among the static obstacles. Right: visualization of the local observation from which the barrier is generated onboard.

Method	Success rate [%]
SDF-MPC	20
DCBF-MPC	40
NTC-MPC	70
ORN-CBF (ours)	100
ON-CBF	100

Table 4: Hardware experiment results for the ground robot over ten randomized obstacle configurations (Dubins car model, MPC horizon

N=10

).

Figure 12: Barrier values of the ORN-CBF and ON-CBF filters in one representative obstacle configuration of the ground-robot hardware experiments, with one run per filter under the same experimental setting and the zero level marked by the dotted line. Briefly negative values are followed by recovery, and no collision occurs. The physical inflation margin separates the represented failure boundary from physical contact.

Figure 1: Effect of the barrier candidate on a CBF-QP safety filter in a dynamic environment, shown at three instants separated by the short interval

\delta

. A ground robot that cannot stop is driven to the right by a goal-directed nominal controller, while an obstacle crosses its path from the front left. Dotted outlines mark the earlier positions, the dashed arrow at each obstacle shows its velocity, and the thin dashed curves show the motion under an extremal admissible input. The red region is the set

\mathcal{X}\setminus\mathcal{S}^{\star}(q)

of robot states from which the collision with an obstacle in state

q

can no longer be avoided at the current heading. Both barriers are evaluated on the relative state

z

of the robot–obstacle pair defined in Section 1 . Left: the barrier is the obstacle distance

c_{\mathrm{rel}}(z)

, whose safe set also contains the red region. That boundary is invisible to the filter and is drawn as a dashed outline only, the filter intervenes too late, and at

t+2\delta

both extremal turns end in a collision. Right: the reachability-based barrier

h^{\star}

is the stationary HJ value function

V_{\infty}(z)

, whose zero level set is the boundary of the red region, so the filter acts already at

t

and the robot stays safe.

Figure 2: Smooth composition of two pairwise barriers along a one-dimensional slice of the state space, for example a straight path from one obstacle to the other. (a) The two pairwise barrier values (blue and orange) form the nonsmooth pointwise minimum (dashed), whose kink at

\bar{x}

marks the change of the active obstacle. The composite barrier

\hat{h}_{\mathrm{cn}}=\mu_{\beta}

(green) runs below that minimum, with a gap that is largest at the kink, where it attains the bound

\ln m/\beta

of Equation 12 . The composite safe interval (green) is slightly smaller than the one given by the exact minimum (gray). (b) The softmax weights Equation 13 along the same slice transfer the gradient continuously from one obstacle to the other instead of switching at

\bar{x}

, and a larger

\beta

narrows this transition.

Figure 3: Structure of the CN-CBF barrier evaluation. The current obstacle states are stacked into a batch, combined with the robot state

x

into the batch of relative states

z_{i}=\sigma(x,q_{i})

, and propagated in parallel through the shared residual network

\hat{r}_{\Theta}

and the closed-form pairwise constraint

c_{\mathrm{rel}}

. Their difference yields the pairwise barrier values of Equation 8 , which the smooth aggregation

\mu_{\beta}

reduces to the composite value in Equation 11 . One reverse-mode differentiation pass through this graph returns all gradients in Equation 15 .

Figure 4: Block diagram of the CN-CBF safety filter. The robot state and the current obstacle-state estimates feed three parallel blocks. The batched CN-CBF evaluation of Figure 3 returns the composite barrier value and its gradients, the robot dynamics model supplies

f_{0}(x)

and

g(x)

, and the backward-difference estimator in Equation 16 supplies the obstacle state derivatives. Together they parameterize the CBF-QP Equation 18 , which minimally modifies the nominal command

u_{\mathrm{nom}}

.

Residual network	MLP, hidden layers $[48,48,24,24,12,12]$ , sinusoidal
Output activation	softplus (nonnegative residual)
Trainable parameters	$4837$
Optimizer / epochs	Adam / $10\,000$
Training time (RTX 3090)	$\sim$ $5.5\text{\,}\mathrm{min}$

Table 1: CN-CBF configuration shared by both robots.

	Ground robot	Quadrotor
Robot model	kinematic unicycle	planar double integrator
Obstacle model	unicycle, bounded turn rate	double integrator
Relative state $z$	Equation 22	Equation 23
HJ grid over $\mathcal{Z}$	$80\times 80\times 20\times 10$	$80\times 80\times 20\times 20$
HJ computation (RTX 3090)	$\sim$ $10\text{\,}\mathrm{s}$	$\sim$ $10\text{\,}\mathrm{s}$

Table 2: CN-CBF configuration specific to each robot.

Figure 5: Positional slice of the learned pairwise barrier

\hat{h}_{\Theta}

for the ground robot, at relative heading

\theta^{\mathrm{rel}}=\pi

(obstacle moving head-on toward the robot) and obstacle speed

v^{\mathrm{o}}=

1.2\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1}$$ , in the robot body frame. The robot is located at the origin and heads along

p^{\mathrm{rel}}_{x}

. The collision set (black circle, zero level of

c_{\mathrm{rel}}

) is the disk of relative obstacle positions within

R_{\min}

of the robot. The numerically computed unsafe region (red shading, zero level of

V_{\infty}

dashed) extends several meters ahead of the robot, because an oncoming obstacle in this region can force a collision on a robot that cannot stop and that turns at a limited rate. The zero level set of the learned barrier (green) closely matches the numerical HJ boundary and, by construction, never enters the collision set.

Figure 6: Ground-robot simulation study over 100 randomized pedestrian-crowd scenarios per density. (a) Success rates of the four MPC-based planners and of the CN-CBF safety filter. The advantage of CN-CBF increases with the crowd density. (b) Path lengths and (c) times to goal of the scenarios completed by all methods. Boxes span the interquartile range, lines mark the median, and whiskers extend to 1.5 times the interquartile range. The higher success rates of CN-CBF are not obtained through detours or overly conservative motion.

Figure 7: Example scene from the ground-robot simulation study. (a) The Gazebo environment with the robot navigating among pedestrians controlled by the social-force model. (b) The corresponding local perception, with detected pedestrians, their estimated velocities, and the planned motion of the robot. (c) Positional slice of the composite barrier

\hat{h}_{\mathrm{cn}}

in the robot frame, with the robot at the origin heading to the right (dart), evaluated over hypothetical robot positions at the current heading, with red shading marking the unsafe positions and arrows showing the pedestrian velocities. The pairwise zero level sets (gray) surround the collision disk of each pedestrian (black) and are elongated along the direction in which that pedestrian moves relative to the robot, while the composite zero level set (green) traces a smooth outline around their union without touching the collision disks.

Figure 8: Quadrotor simulation study over 100 randomized scenarios per density, comparing the CN-CBF filter against the collision-cone CBF (C3BF) and the higher-order CBF (HO-CBF), all combined with the same LQR nominal controller. (a) Success rates. (b) Path lengths and (c) times to goal of the scenarios completed by all methods, with boxes as in Figure 6 . CN-CBF maintains the highest success rates at every density while matching the baselines in path length and motion time.

Figure 9: Composite barrier value

\hat{h}_{\mathrm{cn}}

(green) and distance

\min_{i}c_{\mathrm{rel}}(z_{i})

to the nearest pedestrian (dashed) over time during a successful collision-avoidance run of the CN-CBF filter on the physical delivery robot. The barrier value anticipates the encounter well before the distance becomes small and briefly becomes negative under model mismatch and measurement noise. The physical buffer contained in

R_{\min}

prevents a collision during this interval.

Figure 10: Barrier values and distance to the nearest obstacle over time in the quadrotor hardware experiments with five moving obstacle drones, for (a) the CN-CBF filter, (b) the C3BF filter [ 150 ] , and (c) the HO-CBF filter [ 163 ] , all around the same LQR nominal controller. For the two baselines, which impose one constraint per obstacle, the five pairwise barrier values

h_{1},\ldots,h_{5}

are shown individually, and each HO-CBF barrier is the distance to the corresponding obstacle by definition. Only the CN-CBF filter keeps the distance positive throughout, while both baselines collide (red circles).

Figure 1: Finite-horizon trajectories for the inverted pendulum, generated by (a) model predictive path integral (MPPI) sampling over the full control set and (b) the proposed vertex-restricted beam search. Both methods start from the same 160 random initial states, drawn as circles and colored by the worst constraint value

\min_{k}c(x_{k})

reached along the trajectory, so that red marks a constraint violation. The subsequent states are drawn as small arrows that point in the direction of motion. The black lines mark the constraint boundary and the purple dashed lines the boundary of the numerically computed maximal safe set. The vertex-restricted search keeps the pendulum inside the constraint set from almost every initial state in the numerically computed safe set, while the full-control sampler fails from many of them.

System ( $n_{x}$ , $n_{u}$ )	$\rho_{\mathrm{FS}}$ (%) $\downarrow$			$\rho_{\mathrm{FU}}$ (%) $\downarrow$			$\eta_{\mathrm{eff}}$ $\uparrow$
System ( $n_{x}$ , $n_{u}$ )	ND	FCD	VRCD (ours)	ND	FCD	VRCD (ours)	ND	FCD	VRCD (ours)
Inverted Pendulum (2, 1)	100.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	15.96 $\pm$ 0.53	1.11 $\pm$ 0.30	0.00 $\pm$ 0.00	0.08 $\pm$ 0.00	0.22 $\pm$ 0.00
1D Double Integrator (2, 1)	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	0.95 $\pm$ 0.56	4.01 $\pm$ 0.42	1.18 $\pm$ 0.31	0.41 $\pm$ 0.00	0.39 $\pm$ 0.00	0.41 $\pm$ 0.00
2D Vertical Drone (2, 1)	36.80 $\pm$ 5.39	–	0.00 $\pm$ 0.00	4.73 $\pm$ 6.81	4.72 $\pm$ 6.81	1.88 $\pm$ 0.46	0.35 $\pm$ 0.20	0.00 $\pm$ 0.00	0.58 $\pm$ 0.01
Dubins Car (3, 1)	0.06 $\pm$ 0.04	0.01 $\pm$ 0.01	0.04 $\pm$ 0.01	4.75 $\pm$ 1.28	4.33 $\pm$ 0.31	2.25 $\pm$ 0.28	0.92 $\pm$ 0.00	0.92 $\pm$ 0.00	0.93 $\pm$ 0.00
2D Double Integrator (4, 2)	0.03 $\pm$ 0.01	0.00 $\pm$ 0.00	0.03 $\pm$ 0.01	4.68 $\pm$ 0.25	6.84 $\pm$ 0.30	3.42 $\pm$ 0.22	0.93 $\pm$ 0.00	0.92 $\pm$ 0.00	0.93 $\pm$ 0.00

Table 1: Validation results for the learned barriers on the 15 evaluated systems, comparing the three supervision settings of Section 1 . Reported are the false-safe rate

\rho_{\mathrm{FS}}

, the false-unsafe rate

\rho_{\mathrm{FU}}

, and the effective safe volume

\eta_{\mathrm{eff}}

of Equation 25 , as mean and standard deviation over five training seeds. A dash marks a setting whose predicted safe set was empty for every seed, for which the false-safe rate is undefined, and a missing standard deviation indicates that the false-safe rate is defined for a single seed only. Best mean values in each group are printed in bold. Settings with an empty predicted safe set are not ranked, because their false-unsafe rate reflects only that every state is predicted unsafe.

System	ND	FCD	VRCD (ours)
Inverted Pendulum	1.54 $\pm$ 3.07	33.53 $\pm$ 0.61	96.39 $\pm$ 1.16
1D Double Integrator	98.37 $\pm$ 0.72	94.02 $\pm$ 0.43	98.10 $\pm$ 0.35
2D Vertical Drone	64.09 $\pm$ 32.05	0.00 $\pm$ 0.00	98.48 $\pm$ 0.20
Dubins Car	99.56 $\pm$ 0.15	99.66 $\pm$ 0.04	99.80 $\pm$ 0.03
2D Double Integrator	99.52 $\pm$ 0.03	99.39 $\pm$ 0.02	99.64 $\pm$ 0.02
Kinematic Bicycle	97.71 $\pm$ 0.24	99.65 $\pm$ 0.00	99.59 $\pm$ 0.00

Table 2: Intersection over union (%) between the predicted safe set

\hat{\mathcal{S}}_{\Theta}

and the numerically computed maximal safe set, evaluated on the grid of the numerical solution for the seven systems with at most four states. Entries are mean and standard deviation over five training seeds.

Figure 2: Learned barriers

\hat{V}_{\Theta}

for the inverted pendulum under the three supervision settings, (a) no data (ND), (b) full-control data (FCD), and (c) vertex-restricted control data (VRCD), shown in the

(\theta,\omega)

plane with red shading marking the states predicted to be unsafe. The black lines mark the constraint boundary, the purple dashed lines the boundary of the numerically computed maximal safe set, and the green lines the boundary of the predicted safe set. The physics-informed loss alone (ND) collapses to a trivial constant solution with an empty predicted safe set. Full-control supervision (FCD) produces a barrier that is safe but strongly conservative. Vertex-restricted supervision (VRCD) approximately recovers the maximal safe set while keeping its predicted safe set inside the constraint set.

Figure 3: Mean squared error of the generated labels

V^{K,B}_{\Delta t}

against the numerically computed value function on the inverted pendulum over a uniform lattice of

100\times 100

states, as a function of the beam width

B

for the horizon lengths

K

and the step sizes (a)

\Delta t=

0.05\text{,}\mathrm{s}$$ , (b)

0.1\text{\,}\mathrm{s}

, and (c)

0.2\text{\,}\mathrm{s}

.

Figure 4: Execution time of the vertex-restricted beam search on the inverted pendulum, as a function of the beam width

B

for the horizon lengths

K

. With a single input, the search tree has at most

2^{k}

nodes at depth

k

, so the beam width takes effect only once the tree is wider than the beam, and for long horizons the runtime grows approximately in proportion to

BK

. It does not depend on the step size

\Delta t

.

$\Delta t$ [ $\mathrm{s}$ ]	$\rho_{\mathrm{FS}}$ (%) $\downarrow$	$\rho_{\mathrm{FU}}$ (%) $\downarrow$	$\eta_{\mathrm{eff}}$ $\uparrow$
0.02	3.94 $\pm$ 0.24	2.88 $\pm$ 0.03	0.340 $\pm$ 0.001
0.05	0.63 $\pm$ 0.03	3.12 $\pm$ 0.02	0.354 $\pm$ 0.000
0.1	1.15 $\pm$ 0.06	2.16 $\pm$ 0.02	0.356 $\pm$ 0.000
0.2	1.18 $\pm$ 0.13	3.22 $\pm$ 0.06	0.348 $\pm$ 0.000
0.4	0.51 $\pm$ 0.13	7.02 $\pm$ 0.14	0.317 $\pm$ 0.000

Table 3: Influence of the step size

\Delta t

on the six-degree-of-freedom underwater vehicle. The horizon length

K=40

and the beam width

B=800

are fixed, so the time horizon

K\Delta t

ranges from

0.8\text{\,}\mathrm{s}

to

16\text{\,}\mathrm{s}

. Entries are mean and standard deviation over three training seeds.

Figure 5: Slices of the learned barrier

\hat{V}_{\Theta}

for the six-degree-of-freedom underwater vehicle in the

(p_{y},p_{z})

plane, one for each step size of Table 3 , with red shading marking the states predicted to be unsafe. The black line is the constraint boundary and the green line is the boundary of the predicted safe set. The predicted safe set becomes visibly smaller at the largest step size.

System ( $n_{x}$ , $n_{u}$ )	$\rho_{\mathrm{FS}}$ (%) $\downarrow$			$\rho_{\mathrm{FU}}$ (%) $\downarrow$			$\eta_{\mathrm{eff}}$ $\uparrow$
System ( $n_{x}$ , $n_{u}$ )	ND	FCD	VRCD (ours)	ND	FCD	VRCD (ours)	ND	FCD	VRCD (ours)
Inverted Pendulum (2, 1)	–	1.06 $\pm$ 0.19	6.49 $\pm$ 3.23	0.00 $\pm$ 0.01	14.91 $\pm$ 0.76	0.29 $\pm$ 0.17	0.00 $\pm$ 0.00	0.08 $\pm$ 0.00	0.21 $\pm$ 0.01
1D Double Integrator (2, 1)	–	0.16 $\pm$ 0.07	0.15 $\pm$ 0.07	26.15 $\pm$ 15.13	4.35 $\pm$ 0.28	1.20 $\pm$ 0.10	0.00 $\pm$ 0.00	0.39 $\pm$ 0.00	0.41 $\pm$ 0.00
2D Vertical Drone (2, 1)	–	–	0.14 $\pm$ 0.06	3.95 $\pm$ 6.12	4.06 $\pm$ 7.96	1.80 $\pm$ 0.14	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	0.58 $\pm$ 0.00
Dubins Car (3, 1)	–	0.08 $\pm$ 0.04	0.18 $\pm$ 0.07	81.06 $\pm$ 4.98	4.78 $\pm$ 1.16	2.51 $\pm$ 0.49	0.00 $\pm$ 0.00	0.92 $\pm$ 0.00	0.92 $\pm$ 0.00
2D Double Integrator (4, 2)	–	1.60 $\pm$ 0.55	1.58 $\pm$ 0.83	67.67 $\pm$ 10.10	3.69 $\pm$ 0.39	3.87 $\pm$ 0.43	0.00 $\pm$ 0.00	0.91 $\pm$ 0.01	0.91 $\pm$ 0.01

Table 4: Validation results without the residual architecture, in which the network predicts the value function directly through a linear output instead of a nonnegative residual subtracted from

c

. All other settings and reporting conventions are those of Table 1 . A dash marks a predicted safe set that was empty for every seed.

Figure 6: Sensitivity to the weight

\lambda

of the physics-informed term on the inverted pendulum. (a) Validation metrics. (b) Mean squared error against the numerically computed value function. Curves are means over three seeds and the bands are one standard deviation. In the shaded region the predicted safe set is empty for

\lambda=0.7

, where

\eta_{\mathrm{eff}}

is zero and

\rho_{\mathrm{FS}}

is undefined, and every state predicted safe is unsafe for

\lambda\geq 0.9

.

Figure 7: Hardware experiment with the delivery robot. (a) The recorded robot motion and the tracked pedestrians in the workspace, drawn as circles colored by time. Opaque circles mark the two encounters in which a pedestrian crosses the path of the robot. (b) The composite barrier value

\hat{h}_{\mathrm{cn}}

over the run, with the zero level marked by the dotted line. While no pedestrian is tracked, the barrier is undefined and its value is drawn at

3\text{\,}\mathrm{m}

. (c), (d) The nominal and the filtered acceleration and turning-rate commands. The filter deviates from the nominal command only while a pedestrian is on an intercepting course, and the barrier value becomes slightly negative for a short period while the pedestrians pass the robot.

Figure 8: View from the onboard computer of the delivery robot during a trial, showing the front camera image, a positional slice of the learned barrier, and the output of the perception module.

Figure 1: The five methods organized by the design choices that separate them. The columns of the grid give the controller interface, its rows give the environment class for which the reachability targets are computed, and each box names the learned representation and its conditioning input. The band below the grid holds the third design choice, the source of the supervision, which VertexCBF generates from the vertex structure of the input set instead of taking it from a grid-based solver.

Method	Interface and setting	Learned representation	Offline supervision	Online operation
NTC-MPC ( Chapter 2 )	MPC terminal constraint for a static environment observed locally	Value function regressed directly by a hypernetwork-generated field conditioned on the observed SDF	One stationary grid-based HJ solution per training observation	One hypernetwork pass per planning cycle, then value and gradient queries of the compact field inside the NLP
RNTC-MPC ( Chapter 3 )	MPC terminal constraint for moving obstacles under a nominal motion prediction	Predicted terminal SDF minus a nonnegative residual, conditioned on snapshots of the predicted SDF sequence	One time-varying grid-based HJ solution per training scene	Motion prediction and SDF sequence, one hypernetwork pass per cycle, then queries of the compact field inside the NLP
ORN-CBF ( Chapter 4 )	CBF-QP safety filter for a static environment observed locally	Observed SDF minus a nonnegative residual generated by a hypernetwork conditioned on the observed SDF	One stationary grid-based HJ solution per training observation, cropped to a central positional patch	One hypernetwork pass per observation update, then value and gradient at the current state and one QP per control step
CN-CBF ( Chapter 5 )	CBF-QP safety filter for a varying number of moving obstacles with bounded inputs	Relative obstacle distance minus a nonnegative residual for one robot–obstacle pair, shared over obstacles and combined by a smooth minimum	One robust stationary grid-based HJ solution per robot–obstacle class	One batched pass over the obstacle list and one QP with a single composite constraint per control step
VertexCBF ( Chapter 6 )	CBF-QP safety filter for a fixed known constraint function	Constraint function minus a nonnegative residual, represented by a single fixed network	Finite-horizon search over input vertices at sampled states, combined with a physics-informed loss	Value and gradient at the current state and one QP per control step, composed over obstacles as in CN-CBF when deployed

Table 1: Overview of the five methods, summarizing for each one the controller interface and setting, the learned representation, the source of the reachability supervision, and the computation performed online.

System	$(n_{x},n_{u})$	State box $\mathcal{X}$	Input box $\mathcal{U}$ , parameters
Inverted Pendulum	(2, 1)	$\theta\in[-0.5,0.5]$ , $\omega\in[-1.5,1.5]$	$\tau\in[-3.5,3.5]$ ; $m{=}2.0$ , $l{=}1.0$ , $g{=}9.81$
1D Double Integrator	(2, 1)	$p\in[-1.5,1.5]$ , $v\in[-1.5,1.5]$	$a\in[-0.5,0.5]$
2D Vertical Drone	(2, 1)	$z\in[-0.5,3.5]$ , $v_{z}\in[-4,4]$	$a\in[-1,1]$ ; $K{=}12.0$ , $g{=}9.81$
Dubins Car	(3, 1)	$p_{x},p_{y}\in[-4,4]$ , $\theta\in[-3.14,3.14]$	$\omega\in[-0.5,0.5]$ ; $v{=}1.0$
2D Double Integrator	(4, 2)	$p_{x},p_{y}\in[-4,4]$ , $v_{x},v_{y}\in[-2,2]$	$a_{x},a_{y}\in[-1,1]$
Kinematic Bicycle	(4, 2)	$p_{x},p_{y}\in[-25,25]$ , $\psi\in[-3.14,3.14]$ , $v\in[0,20]$	$\tan\delta\in[-0.3,0.3]$ , $a\in[-3,3]$ ; $L{=}2.8$

System	Constraint function $c(x)$	Parameters
Inverted Pendulum	$d-\lvert\theta\rvert$ (pole-angle bound)	$d{=}0.3$
1D Double Integrator	$d-\lvert p-p_{\mathrm{c}}\rvert$ (position band)	$p_{\mathrm{c}}{=}0.0$ , $d{=}1.0$
2D Vertical Drone	$d-\lvert z-z_{\mathrm{c}}\rvert$ (altitude band)	$z_{\mathrm{c}}{=}1.5$ , $d{=}1.5$
Dubins Car	$\sqrt{p_{x}^{2}+p_{y}^{2}}-r$ (circular obstacle)	$r{=}1.0$
2D Double Integrator	$\sqrt{p_{x}^{2}+p_{y}^{2}}-r$	$r{=}1.0$
Kinematic Bicycle	$\sqrt{p_{x}^{2}+p_{y}^{2}}-r$	$r{=}7.0$

System	Search variant	$N_{\mathrm{data}}$	$K$	$\Delta t$ [ $\mathrm{s}$ ]	$K\Delta t$ [ $\mathrm{s}$ ]	$B$	Extra	$T_{\mathrm{data}}$
Inverted Pendulum	Beam	$3600$ ( $60{\times}60$ )	20	0.10	2.0	$500$	—	$1\text{\,}\mathrm{s}$
1D Double Integrator	Stochastic beam	$2500$ ( $50{\times}50$ )	40	0.10	4.0	$1500$	softmax, $T{=}0.05$	$1\text{\,}\mathrm{s}$
2D Vertical Drone	Branch and bound	$6400$ ( $80{\times}80$ )	60	0.05	3.0	$2000$	2 passes	$5\text{\,}\mathrm{s}$
Dubins Car	Beam	$50\,000$	150	0.05	7.5	$1000$	—	$27\text{\,}\mathrm{s}$
2D Double Integrator	Branch and bound	$300\,000$	100	0.05	5.0	$1000$	3 passes	$8\text{\,}\mathrm{min}$ $2\text{\,}\mathrm{s}$
Kinematic Bicycle	Beam	$200\,000$	80	0.10	8.0	$1000$	—	$2\text{\,}\mathrm{min}$ $24\text{\,}\mathrm{s}$

System	Hidden	$\beta$	$N_{\mathrm{PDE}}$	$\lambda$	LR drop	$T_{\mathrm{train}}$
Inverted Pendulum	$[32,32,32,32,32]$	10	$10\,000$	0.20	$7000$	$1\text{\,}\mathrm{min}$ $0\text{\,}\mathrm{s}$
1D Double Integrator	$[32,32,32,32]$	1	$10\,000$	0.90	$8000$	$51\text{\,}\mathrm{s}$
2D Vertical Drone	$[32,32,32,32]$	10	$10\,000$	0.55	$7000$	$52\text{\,}\mathrm{s}$
Dubins Car	$[32,32,32,32]$	10	$200\,000$	0.80	$8000$	$2\text{\,}\mathrm{min}$ $15\text{\,}\mathrm{s}$
2D Double Integrator	$[32,32,32,32,32]$	10	$600\,000$	norm.	$7000$	$6\text{\,}\mathrm{min}$ $46\text{\,}\mathrm{s}$
Kinematic Bicycle	$[32,64,64,32]$	10	$500\,000$	0.15	$7000$	$6\text{\,}\mathrm{min}$ $3\text{\,}\mathrm{s}$

Learning Reachability-Based Safety Representations for Real-Time Robot Navigation in Unknown and Dynamic Environments

Abstract

Figures & tables

Appendix

Safe Robot Navigation

Gradient-Free Neural Hamilton-Jacobi Reachability for Scalable Safety-Critical Control

HJ-SafeDMP: Hamilton-Jacobi Reachability-Guided Dynamic Movement Primitives for Provably Safe Robot Motion

Contingency-Aware Planning via Certified Neural Hamilton-Jacobi Reachability

System	$N_{\mathrm{valid}}$	$T_{\mathrm{valid}}$ [ $\mathrm{s}$ ]	$\Delta t_{\mathrm{valid}}$ [ $\mathrm{s}$ ]	Wall-clock
Inverted Pendulum	$20\,000$	5.0	0.010	$1\text{\,}\mathrm{s}$
1D Double Integrator	$20\,000$	4.0	0.010	$1\text{\,}\mathrm{s}$
2D Vertical Drone	$20\,000$	5.0	0.010	$1\text{\,}\mathrm{s}$
Dubins Car	$200\,000$	5.0	0.010	$3\text{\,}\mathrm{s}$
2D Double Integrator	$1\,000\,000$	5.0	0.010	$12\text{\,}\mathrm{s}$
Kinematic Bicycle	$500\,000$	5.0	0.010	$6\text{\,}\mathrm{s}$

Abstract

Safe Robot Navigation

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