Energy-Based Safety Methodologies

• Energy-based safety methodologies are control approaches that use energy or barrier functions to partition state space into safe and unsafe regions.
• These methods employ quadratic programs, passivity-based energy tanks, and adaptive filters to enforce real-time safety constraints and balance performance with risk.
• Applications span collaborative robotics, power systems, and autonomous vehicles, ensuring reliable safety guarantees while optimizing efficiency.

Energy-based safety methodologies are a class of control and verification techniques that ensure system trajectories remain within provably safe regimes, as defined by physical or abstract energy-like functions. These methods typically center on constructing scalar energy, potential, or barrier functions whose level sets stratify the state space into safe and unsafe domains, and then design real-time control laws or decision frameworks to guarantee forward invariance of the safe set under the system’s dynamics. Applications span collaborative robotics, power systems, marine navigation, autonomous vehicles, battery systems, and edge-computing for safety-critical AI, and they facilitate trade-offs between efficiency and risk within a mathematically rigorous framework.

1. Foundations of Energy-Based Safety Certification

The foundational paradigm considers a control-affine nonlinear system

$$\dot{x} = f(x) + g(x)u,\quad x\in\mathbb{R}^n,\; u\in\mathbb{R}^m,$$

equipped with a continuously differentiable scalar function $V: \mathbb{R}^n \to \mathbb{R}$ termed a safety function, energy function, or barrier, whose sublevel set $\mathcal{C} := \{x\,|\,V(x)\leq 0\}$ defines safety. Unsafe states correspond to $V(x) > 0$ (Wei et al., 2019).

The control objective is to ensure, under all operating conditions, that the closed-loop vector field never causes $V(x)$ to increase beyond zero. In this framework, safety constraints are enforced by solving a quadratic program (QP) at each state:

$$u^* = \arg\min_{u \in \mathbb{R}^m} \|u-u_{\mathrm{nom}}(x)\|^2 \quad \text{s.t.} \quad L_f V(x) + L_g V(x)\,u + \alpha(V(x)) \leq 0,$$

where $L_f V$, $L_g V$ are Lie derivatives and $\alpha(\cdot)$ is a class-$\mathcal{K}$ function (typically linear). This parametrization admits potential-field, barrier, and safe set controllers as special cases, providing a unified theoretical basis (Wei et al., 2019).

2. Methodological Variants and Unified Frameworks

Key instantiations within this framework include:

• Potential Field Methods (PFM): Use steepest-descent dynamics of a repulsive potential function; activated only when $V(x) \geq 0$.
• Sliding Mode Algorithms (SMA): Apply discontinuous control proportional to the sign of $L_g V(x)$, guaranteeing $\dot{V} < 0$ via high-gain switching.
• Safe Set Algorithms (SSA): Solve the QP with a fixed negative slack, activating only when $V(x) \geq 0$; suitable for systems with known model uncertainties.
• Barrier Function Methods (BFM): Introduce a linear class-$\mathcal{K}$ function in the QP constraint to enforce exponential convergence to the safe set everywhere in state space.

A new hybrid, the Sublevel Safe Set (SSS) algorithm, applies a barrier-type constraint only on the super-level set $\{V\geq0\}$: the QP is solved for $V(x)\geq0$, while the nominal input is used for $V(x)<0$. Closed-form solutions are obtainable via perpendicular decomposition (Wei et al., 2019).

A tabular summary of benchmarked hybrid scores—where higher is better—on canonical systems is as follows:

Method	Ball	Unicycle	SCARA	4 DOF Arm
SSA	5.17	2.37	—	4.53
SMA	5.37	3.23	0.39	5.83
PFM	2.50	—	0.03	4.10
BFM	6.07	2.73	0.37	2.53
SSS	7.23	3.13	0.96	5.23

The SSS typically yields best or near-best performance across benchmarks, indicating the value of integrating barrier and switching ideas in safety design (Wei et al., 2019).

3. Energy-Based Safety in Human–Robot Physical Interaction

In collaborative robotics, "energy-based safety" is codified in standards such as ISO/TS 15066, where the safety concept is realized by limiting the total mechanical energy transferable from the robot to a human during impact (Zanella et al., 27 Jan 2026). The energy-based constraint is derived by modeling transient contact as a lumped mass–spring–mass system, quantifying maximum permissible kinetic energy, and mapping empirical biomechanical pain-onset thresholds (max force or pressure) into elastic energy limits. The core ISO/TS 15066 inequalities are:

$$K_0 \leq K_{0,\max} = \frac{m_R + m_H}{m_H}\,U_{s,\max},\qquad U_{s,\max} = \frac{F_{\max}^2}{2k}.$$

Here, $m_R$ is the robot's effective mass (reflected inertia), $m_H$ is human effective mass, $k$ tissue stiffness, and $F_{\max}$ the allowed contact force. Conservative assumptions (e.g., fully inelastic collisions, large $m_H$) set strict kinetic energy or velocity limits, directly trading off allowable robot performance against guaranteed safety (Zanella et al., 27 Jan 2026).

Design parameters affecting allowable energy include contact area, tissue stiffness modulation, directional reflected inertia minimization (computable in real time), and context-aware switching between transient and quasi-static force limits. Controller architectures to enforce energy-based safety comprise passivity-based energy tanks, impedance control with time-varying bounds, and real-time QPs filtering commands to satisfy $\frac{1}{2}m_R v^2 \leq K_{0,\max}$ (Zanella et al., 27 Jan 2026).

4. Passivity-Based and Barrier-Certified Energy Filters

Energy Tank Framework. The energy tank formalism is a passivity-based enforcement mechanism whereby a virtual dynamical system ("tank") stores and releases energy such that the system kinetic energy never exceeds a prescribed safety budget. The energy tank state $x_t$ encodes stored energy $T(x_t) = \frac{1}{2} x_t^2$. A modulated interconnection ensures the combined robot-plus-tank system is passive, and a real-time optimzer projects the desired input to enforce $H(t) \leq E_{\max}$ at all times (Benzi et al., 2023).

Key architectural features include:

• Convex QP at each cycle to ensure tank energy remains above a lower bound.
• Direct enforcement of ISO/TS 15066 energetic constraints, avoiding over-conservatism from two-body approximations.
• Quantified gains: up to 50% higher peak speeds versus standard approaches while never violating energy bounds.

Limitations include the requirement for accurate velocity sensing and the need for robustification when switching energy budgets rapidly (Benzi et al., 2023).

Advanced Safety Filter for Battery Systems. In power-electronics (e.g., BESS), energy-based safety filters combine a control barrier function (CBF) for hard safety (e.g., over-current limits) and a control Lyapunov function (CLF) for finite-time convergence to nominal reference. Utilizing sum-of-squares programming, polynomial CBF and CLF are synthesized and implemented online as a quadratically constrained QP that filters nominal controls. The CLF guarantees smooth, non-chattering reintegration into unconstrained nominal control after the safety threat is eliminated (Schneeberger et al., 2024).

5. Learning-Based and Data-Driven Energy Safety

Bayesian Energy-Aware CBFs. For systems with data-driven or uncertain dynamics, energy functions may be constructed from a posterior over Hamiltonians learned via Gaussian processes (GPs), particularly in port-Hamiltonian or mechanical systems. The energy-aware Bayesian control barrier function (EB-CBF) methodology constructs conservative energy-based barriers from GP posteriors, providing high-probability guarantees on safety. The EC-CBF constraint incorporates both the posterior mean and credible intervals, and a robust worst-case CBF condition is enforced via a QP safety filter (Leung et al., 30 Dec 2025).

The formulated probability:

$$\mathbb{P}\bigl[x(t)\in\{h_\theta\geq0\}\;\forall t\geq 0\bigr] \geq 1-(\eta_{\mathrm{dyn}}+\eta_{EB}),$$

gives the confidence level that trajectories defined by the robust filter will remain within the (true) energy-safe set for all time (Leung et al., 30 Dec 2025).

6. Multiobjective and Economic Energy-Based Safety

Many real systems require balancing safety with efficiency. For marine vehicles, the economic model predictive control (EMPC) formulation explicitly co-optimizes trajectory safety (e.g., minimal distance to obstacles) and actuator energy consumption. The reference trajectory is precomputed for maximum clearance, and the EMPC cost includes terms for electrical power, tracking error, and actuation effort. The safety–energy trade-off is tunable via a scalar coefficient and is continuously adapted based on mission context or state-of-charge (Liang et al., 2021). Empirical results confirm trade-off curves in real hardware trials.

In safety-aware edge-AI control, a "safety state" is formally defined by the remaining time before critical violation (safe time horizon), which determines scheduling of high-cost (energy-intensive) perception tasks. This enables near-optimal energy savings through adaptive offloading or sensor gating, without ever violating a provable safety envelope enforced by online filtering (Odema et al., 2023).

7. Power-Based and Dynamical Adaptation

Energy-based methods can be further refined by directly limiting power (the rate of energy transfer), rather than only cumulative energy. For mechanical systems such as aerial vehicles, instantaneous power constraints—enforced via dynamically adaptive CBFs—are modulated in response to real-time measurements of system stability. The largest Lyapunov exponent (LLE) of error dynamics is computed online; when LLE is positive (indicating divergence), the allowable power is tightened, forcing dissipation and mitigating instability. The real-time safety layer solves a QP at each control cycle, enforcing power and actuation constraints, and is shown to prevent destabilizing energy surges during both free-flight and physical interaction (Cuniato et al., 2022).

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Energy-based safety methodologies thus constitute a versatile, physically interpretable, and mathematically rigorous toolkit for achieving safety in complex control systems. The spectrum of methods spans classical and modern barrier-based control, passivity and energy tank approaches, certified learning, economic optimization, and power-adaptive protocols, applicable across domains from collaborative robotics to power systems, marine navigation, aerial vehicles, and intelligent embedded platforms (Wei et al., 2019, Zanella et al., 27 Jan 2026, Benzi et al., 2023, Schneeberger et al., 2024, Leung et al., 30 Dec 2025, Liang et al., 2021, Odema et al., 2023, Cuniato et al., 2022).