Data-Driven Hamiltonian for Direct Construction of Safe Set from Trajectory Data

TL;DR

This work addresses safety verification for systems with uncertain dynamics by introducing a data-driven Hamiltonian (DDH) that provides a conservative lower bound on the Hamiltonian $H(x,p)$ using trajectory data and Lipschitz-based uncertainty sets. By solving DDH-based Hamilton-Jacobi reachability problems, the authors compute guaranteed inner-approximations of safe sets and derive safe-control policies without requiring a full dynamics model. They further propose an iterative safe-set expansion framework that safely collects data and updates the safe set, demonstrated on the tiltrotor XV-15 to expand its flight envelope from near-hover to forward flight. The approach integrates data-driven analysis with prior knowledge to enable scalable, conservative safety verification and safe experimentation in real-world systems.

Abstract

In continuous-time optimal control, evaluating the Hamiltonian requires solving a constrained optimization problem using the system's dynamics model. Hamilton-Jacobi reachability analysis for safety verification has demonstrated practical utility only when efficient evaluation of the Hamiltonian over a large state-time grid is possible. In this study, we introduce the concept of a data-driven Hamiltonian (DDH), which circumvents the need for an explicit dynamics model by relying only on mild prior knowledge (e.g., Lipschitz constants), thus enabling the construction of reachable sets directly from trajectory data. Recognizing that the Hamiltonian is the optimal inner product between a given costate and realizable state velocities, the DDH estimates the Hamiltonian using the worst-case realization of the velocity field based on the observed state trajectory data. This formulation ensures a conservative approximation of the true Hamiltonian for uncertain dynamics. The reachable set computed based on the DDH is also ensured to be a conservative approximation of the true reachable set. Next, we propose a data-efficient safe experiment framework for gradual expansion of safe sets using the DDH. This is achieved by iteratively conducting experiments within the computed data-driven safe set and updating the set using newly collected trajectory data. To demonstrate the capabilities of our approach, we showcase its effectiveness in safe flight envelope expansion for a tiltrotor vehicle transitioning from near-hover to forward flight.

Figures (4)

• Figure 1: Illustration of Data-driven Hamiltonian. (left) Velocity space indexed by state. The trajectory data consist of state velocities $v^{i}$ indexed by states $x^{i}$ (blue). The true VFB at a query state $x$, $F(x)$, is unknown (red). We can estimate $F(x)$ by mapping data $v^{i}$ at $x^{i}$ to true velocity $\widetilde{v}^{i}$ at $x$. (right) Velocity space at $x$ (top-down view of the left). $\widetilde{v}^{i}$ lies in an uncertainty set $\mathcal{E}(x;x^{i})$ propagated from $x^{i}$ to $x$ (blue circle). Given the costate $p$, the DDH $\widehat{H}(x,p)$ in \ref{['eq:ddh_general']} takes the best guess $\widehat{v}^{*}$ among $\widehat{v}^{i\circ}$'s, the worst-case realization of $\widetilde{v}^{i} \in \mathcal{E}(x;x^{i})$. This procedure ensures $\widehat{H}(x,p) \leq H(x,p)$.
• Figure 2: Random Polynomial Systems. (a) Example of the vector field bound $F(x)$ at various states and $\mathop{\mathrm{arg\,max}}\limits_{v \in F(x)} p^\top v$, illustrating the nonconvexity of the optimization. (b) Data-driven safe sets ($Avoid$ BRT) computed using the DDH: (1) hyperrectangle DDH using tight sensitivity matrix $L^{\text{io}}$ (blue), (2) $l_2$-ball DDH based on Lipschitz constant $L^{x}$ (red), (3) hyperrectangle DDH with doubled matrix $L^{\text{io}}$ (yellow), and (4) hyperrectangle DDH with fewer data points (purple).
• Figure 3: (a)-(c) NASA-Army-Bell XV-15 tiltrotor aircraft in various rotor configurations (Source: NASA). (d) Free body diagram of XV-15 for its longitudinal dynamics in \ref{['eq:xv15-eom']}.
• Figure 4: XV-15 safe set expansion conducted for 20 iterations under Algorithm \ref{['alg:experiment-design']}, during its flight mode transition from near-hover ($\beta=90^\circ$) to cruise ($\beta=0^\circ$). (a) Data-driven safe sets at various iterations, shown as 2D slices in $\mathrm{v}$-$\gamma$ at various tilt angles $\beta$. (b) The safe sets shown in 3D state space at iterations $k=0,4,19$. At each iteration, the trajectories are collected under the data-driven safety-filtered exploration policy, ensuring that they stay within the previous iteration's safe set. After the safe set is updated based on new data, data reduction is conducted to prune data irrelevant for safe set computation.

Theorems & Definitions (15)

• Remark 1
• Proposition 1
• Remark 2
• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 2
• proof
• Theorem 3: Viscosity solution theorem