Energy-Aware Cartesian Impedance Controller

• The paper introduces an innovative energy tank mechanism that constrains power to guarantee passivity during unpredictable contact transitions.
• It combines classical impedance control with real-time energy scaling to prevent unsafe force spikes and ensure smooth robotic manipulation.
• Empirical results demonstrate up to 95% force reduction and enhanced robustness in reinforcement learning and multi-robot applications.

An energy-aware Cartesian impedance controller is a control scheme that extends classical Cartesian impedance control with explicit energy- or power-related safety constraints, guaranteeing passivity and safe interaction in contact-rich environments. Such controllers are increasingly central in robotic manipulation, collaborative assembly/disassembly, aerial physical interaction, and complex reinforcement learning (RL) tasks where safe and robust behavior during contact transitions and unpredictable disturbances is required. The core innovation is the introduction of energy tanks or energy scaling layers, which modulate the control authority based on on-line physical energy storage, thereby enforcing strict safety bounds regardless of task dynamics or policy outputs.

1. Classical Cartesian Impedance Law and its Limitations

The standard Cartesian impedance controller operates by commanding the robot end-effector to behave as a virtual spring-damper system in Cartesian space. Let $x\in\mathbb{R}^3$, $R\in SO(3)$ denote the end-effector position and orientation, and let $(x_d, R_d)$ denote the desired pose. The controller defines errors

$$e_x = x_d - x, \quad \dot{e}_x = \dot{x}_d - \dot{x}$$

$$e_R = (R_d^\top R - R^\top R_d)^\vee, \quad \omega_e = \omega_d - \omega,$$

where $(\cdot)^\vee$ denotes the vector form of a skew-symmetric matrix. With symmetric positive definite gains $$K_p, K_d, K_{p,R}, K_{d,R} \in \mathbb{R}^{3 \times 3}$$, the Cartesian impedance-control wrench is

$$F = K_p\,e_x + K_d\,\dot{e}_x,\quad \tau = K_{p,R}\,e_R + K_{d,R}\,\omega_e,$$

stacked as $$F_{\mathrm{imp}} = \begin{bmatrix}F \ \tau\end{bmatrix}$$. The commanded joint torques are $\tau_{\mathrm{cmd}} = J(q)^\top F_{\mathrm{imp}}$, with $J(q)$ the manipulator Jacobian (Huang et al., 17 Nov 2025, Larby et al., 2022, Hjorth et al., 2023).

While classical impedance controllers provide compliance and stability, they cannot guarantee safety if energy or power injection is not explicitly bounded—especially during unpredictable contact transitions (loss or engagement) or policy overshoots in learning-based methods. This can result in unsafe impulsive interactions, slamming, or energy dissipation violations.

2. Energy Tank Framework and Passivity Enforcement

The central mechanism enabling "energy awareness" is the introduction of an auxiliary energy tank, a constrained storage variable $\xi$ (or $E_{\mathrm{tank}}$), maintaining a strict accounting of energy available for actuation: $\xi \in [0, E_{\max}]$ The tank is updated at every control step: $$\xi_{t+1} =\min\{E_{\max},\,\max\{0,\,\xi_t - p_t\,\Delta t + U_{\mathrm{in}}\}\}$$ where $p_t$ is the instantaneous mechanical power delivered to the environment, and $U_{\mathrm{in}}$ represents any external (e.g., gravity) energy replenishment (Huang et al., 17 Nov 2025, Hjorth et al., 2023, Brunner et al., 2022). No more energy can be extracted than is stored in the tank, ensuring closed-loop passivity.

The wrench (force/torque command) is then scaled: $$Y_t = \min\left(1, \frac{P_{\max}\, \xi_t + U_{\mathrm{in}}}{\max(\varepsilon, p_t)\, \Delta t}\right),\quad F_{\mathrm{imp}}^{\mathrm{safe}} = Y_t F_{\mathrm{imp}}$$ with $P_{\max}$ a hard power limit and $\varepsilon>0$ for regularization. The resulting command is always physically feasible and cannot drive the system outside its energetically safe envelope.

3. Controller and Safety Layer Integration

The general structure of an energy-aware Cartesian impedance controller combines the classical impedance law with the tank-based safety mechanism and, optionally, dynamic damping and gain scaling:

1. Impedance Law: Compute pose/orientation errors and apply the desired stiffness/damping.
2. Tank and Scaling: Compute the current energy tank level, scale the commanded wrench using the tank constraint, and update tank according to the actual power flow (Huang et al., 17 Nov 2025, Hjorth et al., 2023).
3. Damping Injection: When the instantaneous power through the manipulator surpasses a safe threshold, increase the joint-space damping adaptively:

$$\beta = \begin{cases} 1, & P_{\mathrm{motion}} \leq P_{\mathrm{damp}}\ \frac{P_{\mathrm{motion}} - P_{\mathrm{damp}}}{\dot{q}^\top B_{\mathrm{init}} \dot{q}}, & \textrm{otherwise} \end{cases}$$

This limits rapid transients and potential overshoots (Hjorth et al., 2023).

1. Gain/Impedance Adaptation: Stiffness and damping gains may be adapted online via switching or interpolation methods, e.g., LMI-based synthesis, to balance tracking and passivity requirements (Larby et al., 2022, Ghorbanpour et al., 2021).

4. Extension to Learning and Multi-Robot Systems

Energy-aware impedance controllers can be directly embedded in RL and optimal control pipelines, acting as a robust safety filter irrespective of high-level policy outputs. For example, in (Huang et al., 17 Nov 2025), the controller is integrated with Probabilistic Movement Primitives (ProMPs) combined with Proximal Policy Optimization (PPO):

• The RL state is augmented with the energy tank value and instantaneous power.
• The policy outputs trajectory residuals (modifications), mapped to desired Cartesian poses.
• The energy-tank scales the impedance wrench before applying it, guaranteeing that no policy can drive the manipulator to unsafe contact, regardless of the learned policy's aggressiveness.

This approach is shown to:

• Reduce peak end-effector wrench and jerk by 30–50%;
• Virtually eliminate overload events (energy/power violations) during manipulation;
• Improve success rates in contact-rich tasks;
• Enable sim-to-real transfer without fine-tuning (Huang et al., 17 Nov 2025).

In cooperative multi-robot systems, the energy-optimal impedance control paradigm further introduces optimization over the gain profiles to minimize total energy consumption or maximize regeneration, subject to passivity and actuator constraints (Ghorbanpour et al., 2021).

5. Passivity/Stability Certification

Rigorous passivity arguments and Lyapunov analysis are foundational to all energy-aware designs:

• For the basic tank mechanism, a composite Lyapunov function

$V = U(e_x, e_R) + T(\dot{x}, \omega) + \xi$

yields

$$\dot{V} \leq -\dot{e}_x^\top K_d \dot{e}_x - \omega_e^\top K_{d,R} \omega_e \leq 0,$$

showing strict dissipation and stability under physically reasonable assumptions (e.g., full rank Jacobian) (Huang et al., 17 Nov 2025, Hjorth et al., 2023).

• The storage-based analysis extends to systems with time-varying gains and underactuation, provided key passivity inequalities hold at all times (Ghorbanpour et al., 2021, Larby et al., 2022).
• For advanced valve policies (aerial systems), the tank's energy is regulated per-port, and global or weighted gain scaling is used to avoid chattering and maintain smooth safe interaction (Brunner et al., 2022).

6. Experimental Evidence and Performance Assessment

Energy-aware Cartesian impedance controllers have demonstrated robust empirical performance across a range of testbeds:

Controller Type	Impact Force Reduction	Compliance	Passivity in Contact Loss
Standard Impedance	Baseline	Moderate	No
Hybrid Force–Impedance (bounded tank)	+Force tracking	Good	Partial (can "slam" at loss)
Energy-Aware Cartesian Impedance	Up to 95%	Enhanced	Guaranteed for all transitions

• In collaborative disassembly with human partners (unscrewing with contact losses), impact force dropped from 15–34 N to ≈1–2.6 N, a greater than 90% reduction, with smooth compliant displacement and robust recovery (Hjorth et al., 2023).
• In contact-rich RL manipulation, energy-aware controllers delivered higher reliability and success (e.g., in maze sliding tasks), smoother sliding motion on unseen surfaces, and robust transfer from simulated to real robots (Huang et al., 17 Nov 2025).
• For aerial robots, tank-regulated impedance maintained stability even in poorly modeled dynamic environments, supporting robust cart-pushing and obstacle crossing despite uncertainties (Brunner et al., 2022).

7. Design, Tuning, and Future Perspectives

Key algorithmic procedures for energy-aware impedance controller design involve:

1. Selection of Cartesian ports and desired virtual elements (springs/dampers).
2. Synthesis and tuning of virtual gains for desired trade-off between global passivity and local performance, employing port-Hamiltonian and linear matrix inequality (LMI) techniques as needed (Larby et al., 2022).
3. Optimization of gain profiles for minimum energy draw or maximum regeneration in multi-manipulator settings (Ghorbanpour et al., 2021).
4. Integration with high-level planning (RMPs, RL) via explicit MDP augmentation or reward modification to penalize unsafe energy flows (Huang et al., 17 Nov 2025).

Notable open directions include:

• Extension to high-DOF, redundantly-actuated, and underactuated systems.
• Further fusion with data-driven policy learning and robust adaptation methods.
• Application to dynamic human-robot collaboration scenarios and safety-critical environments where formal energetic envelopes are mandated.

Energy-aware Cartesian impedance controllers represent a theoretically principled and practically validated solution for ensuring safety and robustness in a wide array of robotic manipulation tasks—with clear analytical guarantees and significant empirical gains over classical controllers (Huang et al., 17 Nov 2025, Larby et al., 2022, Hjorth et al., 2023, Brunner et al., 2022, Ghorbanpour et al., 2021).