A Reachability-based Safety Certificate for Dynamical System Motion Policies

Aditya Vats; Tianyi Xia; Nadia Figueroa

cs.ROSep 29, 2026

A Reachability-based Safety Certificate for Dynamical System Motion Policies

Authors: Aditya Vats, Tianyi Xia, Nadia Figueroa

Organizations: The authors are with the University of Pennsylvania, Philadelphia, PA, USA.

Abstract

Dynamical Systems (DS) are reactive motion policies representing vector fields trained with theoretical guarantees of stability and convergence. To ensure safety during deployment in unknown environments they must be locally reshaped, either through modulation or geometric control barrier function strategies. However, depending on the geometry of the obstacles and the complexity of the DS, these local strategies can lead the system to unavoidable collisions or spurious attractors. In this work, we certify safety with a value function drawn from the notion of backward reachability tube, which measures the worst-case safety along a rollout trajectory of the nominal DS. Usually, such a value function is intractable for a controlled system due to curse of dimensionality. We show that in the DS-based learning-from-demonstration setting, the absence of a control input collapses the reachability problem to a deterministic rollout, and the presence of certain stability conditions truncates the infinite horizon to a finite one, resulting in a well-defined value function. We further show that the value function we devised is the maximal forward-invariant subset of the obstaclefree region for the nominal DS flow. The application of this certificate function is validated across five DS constructions - analytical, Neural ODE, diffeomorphic latent space, LPV-DS, SE(3)and validate it on a Franka manipulator. Modulation and geometric CBFs also suffer from saddle point in cases of headon approach towards an unsafe zone. We show that CBF-on-V avoids this pitfall entirely.

Figures & tables

Fig. 1 : Head-on behavior on a star obstacle. Background shading is the safety margin $h=\Gamma-1$ (red: obstacle interior, blue: free space, opacity: distance to the boundary). Dots are far initial conditions, diamonds are on-obstacle initial conditions on the head-on arc, $\times$ marks the spurious equilibrium, and the star is the goal; black/teal trajectories reach the goal, red trajectories are trapped. (a) Reference modulation: the head-on line and one surface seed fall into a saddle . (b) CBF-on- $h$ : a spurious attractor whose basin traps the far starts and the whole notch arc of surface seeds. (c) Stacked CBF-on- $V$ + hard- $h$ (ours): every start, on-obstacle or far, is steered around to the goal while keeping $h\geq 0$ .

Panel of Figure 2 — Fig. 2 : Main results across the four DS constructions. In each panel the nominal (dashed) and filtered (solid) rollouts are overlaid on the reachability value landscape $V$ , with the obstacle marked; only the computation of $V$ changes between panels, the filter (Eq. ( 12 )) is identical. (a) Spiral DS with closed-form $V$ : the nominal arc enters the obstacle, the filtered rollout rides the $\{V=0\}$ boundary and converges. (b) Neural-ODE DS with a learned value network on a LASA shape (JShape): four initial conditions shown, three with $V(x_{0})<0$ and one with $V(x_{0})\geq 0$ , each nominal/filtered pair in the same colour. (c) LPV-DS with the certified exponential rate of $\alpha_{L}=\min_{k}\lambda_{\min}(M_{k})\,/\,\lambda_{\max}(P)$ as noted in section V-D . (d)-(e) Diffeomorphic latent DS: the certificate is enforced in latent space (d) and the same four rollouts are shown pulled back through $\psi^{-1}$ into task space (e).

Fig. 3 : Three filters on the spiral DS from the same initial condition over the same obstacle acting on a nominally safe trajectory that grazes by the obstacle. CBF-on- $V$ (ours) coincides with the nominal, while CBF-on- $h$ and modulation deflect a trajectory that was already safe. Bottom: the three correction magnitudes $\lVert u(t)\rVert$ for the rollout sequence.

Metric	CBF-on- $V$ (ours)	CBF-on- $h$	mod.
Intervention rate	$\mathbf{0}$ [2pt] (never activates)	0.041	0.278
Total effort		0.288	0.300
Peak correction		0.552	0.492
Path deviation		0.692	0.365

TABLE I : Filter behaviour on the spiral DS. They differ in intervention. CBF-on-

V

never activates as nominal trajectory is already safe. Metrics: intervention rate

=\#\{\|u\|>\epsilon\}/N

; total effort

=\int\|u\|\,dt

; peak

=\max_{t}\|u\|

; path deviation

=\int\|x_{\mathrm{filt}}-x_{\mathrm{nom}}\|\,dt

;

Panel of Figure 5 — Fig. 4 : Reactive avoidance of a hand-moved obstacle on the Franka for two SE(3) skills. Top: video frames at $t=0\text{--}3$ . Bottom: nominal (grey dashed) and filtered end-effector paths, the latter coloured by the task-space correction $\lVert u\rVert$ ; the mocap-tracked obstacle is a time-graded red trail, solid at closest approach.

	Bottle-to-shelf		Pouring		Plate std.-to-lying
Metric	ours	CBF- $h$	ours	CBF- $h$	ours	CBF- $h$
Intervention lead (s) $\uparrow$	2.12	0.11	1.11	0.12	0.86	0.12
Correction-active fraction $\downarrow$	0.067	0.095	0.239	0.322	0.205	0.139
Boundary distance (cm) $\downarrow$	13.6	34.8	34.4	42.4	20.9	33.3
Command jerk RMS (m/s 3 ) $\downarrow$	163	460	281	1104	85	92

TABLE II : SE(3) hardware: reachability filter (CBF-on-

V

, ours) vs. geometric distance CBF (CBF-on-

h

) on the same learned policy per task; no collision in any run. Arrows give the better direction; best per task/metric in bold.

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The goal of this paper is certifying safety of dynamical systems subject to uncertainty. Existing approaches use trajectory data to estimate transition probabilities, and compute safety probabilities recursively via dynamic programming (DP). This recursion may lead to compounding errors in the certified safety probability, thus collapsing to a vacuous lower bound for growing horizons

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Abstract

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