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A Reachability-based Safety Certificate for Dynamical System Motion Policies

Aditya Vats, Tianyi Xia, Nadia Figueroa

arXiv:2609.38421v1cs.RO

TL;DR

Dynamical systems learned from demonstrations need obstacle-safety handling, while local modulation and geometric barrier methods can encounter undesirable equilibria. This work certifies safety along nominal rollouts using a reachability-based value function simplified by autonomous dynamics and finite-time convergence. The paper establishes a maximal forward-invariant safe set and reports that its certificate avoids a head-on stagnation failure of the compared methods.

  • Problem

    Dynamical systems learned from demonstrations lack native obstacle handling, and local modulation or geometric-CBF corrections can induce saddle points, spurious attractors, or limit cycles.

    Source: §I · p. 1

  • Method

    The paper defines a reachability-based safety value over nominal DS rollouts, using autonomous dynamics for deterministic evaluation and stability conditions for finite-horizon truncation.

    Source: §I · pp. 1–2, B. Finite-Horizon Truncation · p. 3

  • Results

    The safe set is the maximal forward-invariant subset of the obstacle-free region under the nominal flow, and the certificate removes the head-on stagnation failure of modulation and geometric-CBF avoidance.

    Source: §I · p. 2, B. Saddle Point Removal w. Stacked… · p. 5

  • Takeaways & Limitations

    The framework is applied across multiple DS formulations, including SE(3), and its conclusion identifies high-dimensional SE(3) implementation as evidence of practicality.

    Source: §VIII · p. 10

Abstract

from arXiv · show

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.

I. INTRODUCTION

Dynamical systems (DS) provide stable motion policies learned from demonstrations, but they have no native obstacle handling, and local safety corrections can impair execution. This work uses a reachability-based value function to certify safety along nominal trajectories, with a deterministic rollout and finite-horizon computation under stated stability conditions.

I. INTRODUCTION

  • For an autonomous DS, the backward-reachability value function becomes the minimum safety value along a deterministic rollout, avoiding the optimal-control term of general reachability.The paper notes that controlled-system reachability requires solving a Hamilton–Jacobi PDE and suffers from the curse of dimensionality.Source: §I · pp. 1–2
  • A known convergence rate yields a closed-form finite-horizon truncation, and forward invariance over that horizon implies invariance for all time.The construction assumes a finite, computable convergence time and a failure region separated from the equilibrium.Source: §I · pp. 1–2
  • The resulting safe set is the maximal forward-invariant subset of the obstacle-free region under the nominal DS flow, making it the least conservative such certificate.The value function is evaluated by integration when the rollout is available analytically, or by a learned approximation otherwise.Source: §I · p. 2
  • The proposed certificate removes a head-on stagnation failure shared by modulation and geometric-CBF avoidance, where it can appear as a saddle equilibrium or spurious attractor.The paper also identifies local methods as potentially inducing limit cycles in complex obstacle geometries or with highly nonlinear DS.Source: §I · pp. 1–2
  • The certificate is instantiated across five DS formulations and validated on a 7-DoF Franka manipulator.The framework handles different representations with the same filter, changing only the value-function computation.Source: §I · pp. 1–2

A. Dynamical Systems (DS)

  • A dynamical system (DS) is a first-order autonomous system whose vector field generates trajectories intended to imitate demonstrations.The paper describes DS policies as robust to perturbations and, when stability is guaranteed, convergent to an equilibrium target.Source: §I · p. 1, A. Dynamical Systems (DS) · p. 2

B. Lyapunov Conditions for Stability

The section establishes how stability and barrier conditions certify forward invariance, then relates backward reachability to a value function whose superlevel set captures safe states.

B. Lyapunov Conditions for Stability

  • A Lyapunov function has zero value at the equilibrium, is positive elsewhere, and does not increase along trajectories; strict decrease certifies convergence to the equilibrium.Exponential stability additionally provides a convergence rate, yielding a finite time to reach any target tolerance.Source: B. Lyapunov Conditions for Stability · p. 2

C. Forward Invariance and Control Barrier Functions

  • Forward invariance means every trajectory starting in a set remains there for all time, so a set excluding the failure region is safe.A barrier function certifies invariance through a pointwise condition on its rate of change along the flow.Source: C. Forward Invariance and Control… · p. 2

D. Backward Reachability and the Value Function

  • The backward-reachability tube identifies states from which the system inevitably enters the failure region despite admissible control, while its value function’s nonnegative superlevel set is safe.Its superlevel set is the maximal control-invariant subset of the safe region; computing the classical Hamilton–Jacobi solution faces a curse of dimensionality.Source: D. Backward Reachability and the Value… · pp. 2–3

III. METHODOLOGY

The method applies backward-reachability certification to autonomous dynamical-system policies under convergence and safety assumptions. Autonomy reduces the value function to a rollout minimum, and finite-time convergence permits a finite horizon for safety certification.

III. METHODOLOGY

  • The method assumes an autonomous DS with a single convergent equilibrium, a computable finite time to enter an η-ball, and a failure region outside that ball.The paper states these assumptions generally hold for a broad class of stable DS learned from demonstration.Source: §III · p. 3

A. Autonomy Collapse of the Value Function

  • With no control input, the reachability optimization reduces to minimizing the safety function h along the deterministic nominal rollout.The resulting stationary value function gives the worst safety value over the trajectory’s entire future, and its nonnegative set is exactly the states whose rollouts never enter failure.Source: A. Autonomy Collapse of the Value… · p. 3

B. Finite-Horizon Truncation

  • A finite, computable convergence time replaces the infinite rollout with a finite one that is equivalent for certifying safety.The truncation relies on a time after which the trajectory has entered and remains within an η-ball about the equilibrium.Source: B. Finite-Horizon Truncation · p. 3

1) Truncation time:

  • The truncation time can come from an exponential-stability rate or from construction of the policy, as in the diffeomorphic latent-space DS.For exponential stability, the Lyapunov convergence rate and bounds yield a closed-form horizon for reaching a target tolerance; the latent-space policy inherits convergence from a stable template.Source: 1) Truncation time · p. 3

2) Equivalence of the truncated value function:

Under Assumptions 1–3, a finite rollout suffices to certify safety for all time. The resulting safe set is the maximal forward-invariant subset of the obstacle-free region under the nominal flow.

2) Equivalence of the truncated value function:

C. Maximality of the Safe Set

  • Under Assumptions 1–3, {x : V(x) ≥ 0} is the maximal forward-invariant subset of the obstacle-free region under the nominal DS flow.It is forward invariant, contained in the obstacle-free region, and contains every forward-invariant subset of that region.Source: C. Maximality of the Safe Set · p. 4
  • The gap between the obstacle-free region and {V ≥ 0} consists of states whose nominal trajectories eventually enter the failure set.These states are instantaneously collision-free, but no forward-invariant subset of the obstacle-free region can extend beyond {V ≥ 0}.Source: C. Maximality of the Safe Set · p. 4
  • Any correct certificate enforcing forward invariance of its safe set must be contained in {V ≥ 0}, making this certificate least conservative.A barrier on h may react to proximity and admit states that later collide, while also influencing states that were never at risk.Source: C. Maximality of the Safe Set · p. 4

D. The Safety Filter

The safety filter uses V to constrain a virtual input, remaining inactive on safe nominal trajectories and intervening when needed to maintain or recover safety.

D. The Safety Filter

  • A minimum-norm virtual input enforces the barrier condition ˙V ≥ −αV V when a state is unsafe or a perturbation pushes it toward ∂SV.For V(x) < 0, the correction drives V upward toward the safe set; near ∂SV, it preserves invariance.Source: D. The Safety Filter · p. 4
  • When V(x) > 0 along a nominal trajectory, V is non-decreasing, so the filter is inactive even when the trajectory passes close to an obstacle.The barrier constraint holds with u = 0 over the entire safe set under nominal DS flow.Source: D. The Safety Filter · p. 4
  • When the DS construction does not guarantee stability, a Lyapunov constraint is added with slack so that safety takes precedence.The Lyapunov and reachability-value barriers use separate gains.Source: D. The Safety Filter · p. 4

IV. OBSTACLE-AVOIDANCE FOR CONCAVE OBSTACLES

For the illustrated star-obstacle head-on case, the reachability-based filter steers every tested start to the goal while maintaining h ≥ 0. Modulation and geometric CBF approaches can stagnate or become trapped in this setting.

A. Modulation and CBF as DS Safety Filters

  • The section compares modulation and geometric CBF filters as local obstacle-avoidance strategies, which can suffer from saddles or undesirable equilibria.The paper reports this issue for both convex and concave geometries and says its reachability-based filter alleviates it.Source: §IV · p. 4, A. Modulation and CBF as DS Safety… · p. 4

IV. OBSTACLE-AVOIDANCE FOR CONCAVE OBSTACLES

  • Every far or on-obstacle start reaches the goal under stacked CBF-on-V + hard-h, while keeping h ≥ 0.In the same star-obstacle comparison, reference modulation has head-on starts falling into a saddle, and CBF-on-h traps far starts and the notch arc of surface seeds.Source: §IV · p. 4, Figure 1 · p. 5

1) Modulation:

  • Modulation compresses the normal and stretches the tangential component to avoid crossing the obstacle boundary.For head-on collisions, f ∥ n makes the velocity zero at the boundary collision point, producing a saddle.Source: 1) Modulation · p. 5

2) Geometric CBF:

  • Geometric CBFs project away the inward normal component; at a head-on boundary collision, this cancels the velocity entirely.Unlike modulation, the barrier does not inject a tangential escape velocity, so the stagnation point can act as a stable attractor.Source: 2) Geometric CBF · p. 5

B. Saddle Point Removal w. Stacked CBF-on-V + CBF-on-h

The stacked filter combines an instantaneous obstacle constraint with a reachability-value constraint to address failures of myopic safety filtering. A side-fixed tie-break can make safe velocity discontinuous across a set of states, but trajectories do not stay on it and convergence is unaffected.

B. Saddle Point Removal w. Stacked CBF-on-V + CBF-on-h

  • The stacked filter imposes one constraint to keep the state outside the obstacle and another to keep its future rollout safe.The first is the instantaneous obstacle safeguard; the second acts on the reachability value.Source: B. Saddle Point Removal w. Stacked… · p. 5
  • At a boundary stall, the reachability constraint makes zero correction infeasible, so the state escapes tangentially rather than stalling.At h = 0 with inward nominal motion, V < 0; the constraints block inward motion while requiring V to rise.Source: B. Saddle Point Removal w. Stacked… · p. 5
  • When two obstacle-clearing sides minimize safety equally, fixing one side makes safe velocity discontinuous across Σ, without affecting convergence.Trajectories do not stay on Σ.Source: B. Saddle Point Removal w. Stacked… · p. 5

V. APPLICATIONS TO DIFFERENT DS CONSTRUCTIONS

  • The certificate is implemented across analytical, Neural ODE, diffeomorphic latent-space, LPV-DS, and SE(3) constructions, with a manipulator deployment.The application sequence moves from closed-form analytical DS to learned and latent policies, then to SE(3) hardware.Source: §V · pp. 5–6

1) Linear DS:

The analytical examples provide closed-form certificates for linear and spiral dynamics, while the Neural ODE example learns the value function when the rollout has no closed form.

2) Spiral DS:

  • For linear and spiral DS, closed-form V and ∇x0V let the filter produce safe velocities directly, correcting initially and then switching off once the nominal flow avoids the obstacle.The spiral DS uses a stable spiral flow and a signed-distance safety function; the reported trajectories illustrate minimum intervention.Source: 2) Spiral DS · p. 6, 1) Linear DS · p. 6

B. Neural ODE

  • For a Neural ODE with no closed-form rollout, an MLP learns V from rollout safety-minimum targets, made tractable by exponential stability.The DS is learned jointly with an ICNN Lyapunov function; the value network and filtered trajectories are shown on the LASA Jshape example.Source: B. Neural ODE · p. 6

C. Latent-Space DS

The certificate extends to diffeomorphic latent-space and LPV dynamics by evaluating safety along their stable flows and applying the filter in the corresponding representation.

C. Latent-Space DS

  • For a diffeomorphic latent-space DS, the certificate is enforced on the stable latent flow, then mapped back so the corrected trajectory clears the real obstacle and converges.The obstacle is represented by an enclosing circle fitted to transformed boundary samples, and the latent value is a closed-form signed distance from the trajectory line.Source: C. Latent-Space DS · p. 6

D. LPV-DS

  • For LPV-DS, a common quadratic Lyapunov function supplies an exponential stability rate, while numerical rollout integration yields V and the filtered safe velocity.The rate is αL = mink λmin(Mk) / λmax(P), and the finite horizon follows from the stability bound.Source: D. LPV-DS · pp. 6–7, Figure 2 · p. 7

E. SE(3) Reachability

The paper extends reachability certification to stable SE(3) LieFlow dynamics on a Franka manipulator and compares the filter with local safeguards. Across the reported spiral-DS comparison, CBF-on-V leaves a nominally safe trajectory unchanged while CBF-on-h and modulation intervene.

E. SE(3) Reachability

  • For SE(3), the certificate evaluates end-effector clearance at rollout samples, smooths the worst rollout clearance with a differentiable softmin, and filters in latent space before mapping the command to a pose.The certificate is deployed on a Franka Research 3, building on a stable Lie-group vector field.Source: E. SE(3) Reachability · p. 7

VI. RESULTS

  • The evaluation spans analytical, learned, and latent dynamics, compares the reachability filter with CBF and modulation, and applies SE(3) LieFlow dynamics to a Franka manipulator.The hardware application is described as demonstrating effectiveness on the CLFD dataset.Source: §VI · p. 7

A. Generality across constructions

  • On the Neural-ODE and diffeomorphic latent examples, filtered rollouts respect the value boundary and clear the obstacle after mapping back to observation space.For the latent example, the certificate is enforced in latent space and the corrected trajectory converges after being pulled back.Source: A. Generality across constructions · pp. 7–8

B. Minimum intervention versus local safeguards

  • In the spiral-DS grazing test, CBF-on-V stays inactive, CBF-on-h briefly perturbs near the obstacle, and modulation corrects from farther away for longer.The comparison uses one initial condition and one obstacle; the correction traces show zero, impulsive, and sustained intervention, respectively.Source: B. Minimum intervention versus local… · p. 8, Figure 3 · p. 8

VII. SE(3) LIEFLOW ON THE FRANKA ARM

The reachability filter transfers to Franka SE(3) skills without policy retraining and is evaluated against a distance CBF and under a moving obstacle. It intervenes earlier and more lightly while avoiding contact and preserving goal tracking.

A. Deployment

  • The Franka deployment applies the unchanged reachability filter to three learned, stable SE(3) skills, adding obstacle avoidance at deployment without retraining the policies.The skills are pouring, bottle-to-shelf, and plate-standing-to-lying; the filtered velocity is applied through a 100 Hz impedance controller.Source: §VII · p. 8, A. Deployment · p. 8

B. Anticipatory versus reactive intervention

  • The reachability filter corrects 0.9–2.1 s before closest approach, versus about 0.1 s for the distance CBF, and yields shorter obstacle-adjacent arcs and lower command jerk across tasks.Neither method collides; Table II compares intervention lead, active fraction, boundary distance, and RMS command jerk per task.Source: B. Anticipatory versus reactive… · p. 8, Table II · p. 9

C. Reactive avoidance under a moving obstacle

  • For the moving-obstacle test, the filter updates the value and its gradient at each control tick using the measured pose and live obstacle centre, then replans avoidance online.The failure region is non-stationary because the hand-held sphere moves during execution.Source: C. Reactive avoidance under a moving… · p. 9
  • With a hand-moved obstacle, the filter's correction stays near zero over most of the path and rises briefly to steer around it; both skills reach their goal poses without contact.Figure 4 depicts the nominal and filtered paths, correction magnitude, and the motion-captured obstacle trail.Source: C. Reactive avoidance under a moving… · p. 9, Figure 4 · p. 9

VIII. CONCLUSION

  • The paper concludes that the reachability approach characterizes a maximal invariant set and supports high-dimensional SE(3) use, while extension to multiple objects remains future work.Source: §VIII · p. 10
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