LASEM: Lightweight Actuation-Space Energy Modeling

• LASEM is a unified modeling approach that formulates compact, actuation-space reduced-order models for diverse robotic and energy systems.
• It employs analytic energy minimization and modal reduction techniques to derive computationally efficient forward and inverse models for applications like soft pneumatic actuators and continuum robots.
• Data-driven corrections integrated with physics-based formulations ensure high prediction accuracy and sub-millisecond evaluation speeds for robust real-time control.

Lightweight Actuation-Space Energy Modeling (LASEM) defines a unified approach to modeling, control, and design of actuators and energy systems by formulating reduced-order models directly in actuation coordinates (forces, pressures, cable displacements, switching rates), rather than relying on full field variables or complex distributed-parameter physics. This paradigm enables analytically compact, computationally efficient forward and inverse models for a wide range of robotic and energy systems, including soft pneumatic actuators, cable-driven continuum robots, serial manipulators, and populations of thermostatic loads. The key principle is to encode all energetic and work-conjugate effects in a functional over the actuation space, allowing for systematic derivation, fast evaluation, and data-driven model correction.

1. Fundamental Principles and Mathematical Structure

LASEM frameworks formalize system energetics by writing the system’s total (pseudo-)potential energy or action functional as a function of explicit actuation-space variables. This eliminates unnecessary internal states while retaining the essential energetic couplings relevant at the control or planning layer.

For example, in the case of cable-driven continuum robots, the actuation-space functional is: $$\Pi[\theta(\cdot);q] = \int_0^L \frac{1}{2}E\,I(s)\,[\theta'(s)]^2\,ds - \sum_{i=1}^2 F_i\,\Delta l_i(\theta)$$ where $\theta(s)$ is the bending angle, $F_i$ the cable forces, $\Delta l_i(\theta)$ the corresponding cable elongations as functionals of the deformation field, $E$ the Young’s modulus, and $I(s)$ the bending stiffness profile (Wu et al., 4 Sep 2025).

For axisymmetric soft pneumatic actuators, the energy functional is decomposed as: $$E(p,h,r_0) = U_{\rm elastic}(\lambda_1, \lambda_2) + U_{\rm press}(p,h,r_0) - W_{\rm ext}(h,F)$$ where $p$ is the pressure, $h$ the controlled height, and $F$ the external lifting force. $(\lambda_1,\lambda_2)$ are the principal stretches, and $U_{\rm elastic}$ is often specified by a hyperelastic model such as Gent (Campbell et al., 1 Apr 2025).

For populations of switching loads (e.g., thermostatic devices), the approach models the aggregate state as a bilinear, reduced-dimensional linear system of Fokker-Planck type in actuation coordinates (the switching rate broadcast signal) (Totu et al., 2014).

This principle generalizes: by writing energetic or dynamical relationships in terms of actuation variables, the complexity of downstream computation and control is minimized.

2. Reduced-Order Model Derivation and Computational Workflows

LASEM employs analytic or semi-analytic solution procedures embedded in a systematic workflow:

• Energy minimization and stationarity: The system equilibrium is characterized via the stationarity of the actuation-space energy, e.g., enforcing $\delta\Pi=0$ in continuum mechanics or $\delta E=0$ in elastomeric actuators. This yields ordinary or partial differential equations for the minimal set of actuation variables (forces, displacements, pressures, etc.) subject to boundary and compatibility conditions.
• Modal or tabular model reduction: Spatial fields are represented in a reduced basis (e.g., constant curvature or modal Galerkin expansions), so that the control-relevant input–output maps (e.g., $F(p,h)$ for pneumatic actuators or $x(L)$ for continuum backbones) may be precomputed, tabulated as splines, or parameterized as low-order polynomials for rapid lookup (Campbell et al., 1 Apr 2025, Wu et al., 4 Sep 2025, Yang et al., 15 Dec 2025).
• Data-driven correction: When pure physics-based models encounter significant empirical discrepancies (as quantified by, e.g., up to 6 N RMSE in SPA force prediction), active learning pipelines are used to train a correction model. This employs ensembles of shallow neural networks (Randomized-Prior-Network, RPN) to learn residuals as a function of geometric and control variables, tightly coupled with active data acquisition strategies to maximize epistemic coverage (Campbell et al., 1 Apr 2025).
• Surrogate regression for manipulators: For serial robots, LASEM instantiates as a regressor-based energy model, where the total electrical power is written as a linear function of features (joint accelerations, currents, velocities), and parameters are estimated via least-squares from operational data, yielding closed-form evaluation of power and current at each timestep (Heredia et al., 8 Aug 2025).

The net computational cost is kept $O(1)$ per evaluation via model distillation, resulting in prediction latencies below 1 ms per query for most applications.

3. Application Domains

LASEM has been instantiated across a diversity of actuation and energy systems:

• Soft Pneumatic Actuators (SPAs): Accurately models the nonlinear force–pressure–height relationships in axisymmetric, strain-limited SPAs, with extensions for automated active-learning-driven design optimization. The approach achieves fast runtime prediction through tabulation/spline compression and empirical correction, supporting geometry optimization for lifting tasks (Campbell et al., 1 Apr 2025).
• Cable-Driven Continuum Robots (CDCRs): Provides both kinematic and dynamic models by minimizing the actuation-space energy, yielding analytic forward/inverse relationships between cable inputs and tip pose or trajectory. The framework includes extensions to nonuniform geometries, cable routing, external distributed loading (e.g., gravity), and axial extensibility (Wu et al., 4 Sep 2025, Yang et al., 15 Dec 2025).
• Serial Manipulators: As realized in the EcBot MATLAB library, LASEM supports data-driven modeling of electrical energy consumption as an actuation-space regressor, from which joint-space trajectory optimization for energy-aware planning is readily performed (Heredia et al., 8 Aug 2025).
• Populations of Thermostatic Loads: Captures the energy aggregate of large populations under actuation via switching-rate broadcast, leading naturally to bilinear Fokker-Planck or PDE models that reduce to low-dimensional ODEs for rapid real-time control use (Totu et al., 2014).

4. Generalization to Novel Geometries and Actuation Modes

LASEM is structurally agnostic to the physical realization of actuation, provided appropriate energetic or stochastic functionals are constructed:

• For continuum robots, it extends from classic constant-curvature cases to robots with nonuniform stiffness, arbirtrary cable routing, and distributed load, only requiring the energetic terms in the functional $\Pi$ to be tailored accordingly (Wu et al., 4 Sep 2025, Yang et al., 15 Dec 2025).
• In pneumatic and soft actuators, the hyperelastic energy function and kinematic mapping may be swapped for different membrane architectures or material models, with the equilibrium equations re-derived by enforcing $\delta E = 0$ for the new layout (Campbell et al., 1 Apr 2025).
• For loads with hybrid/discrete states, only the mode-dependent drift and noise functions need to be restated in the single-device SDE, and grid-level models reconstructed via the same forward–Kolmogorov projection and ODE reduction (Totu et al., 2014).

The transferability of key formulas and the possibility for application-specific compression/approximation strategies (e.g., splines, polynomial fitting, basis function expansions, quantized encoding) underscore the adaptability of the LASEM philosophy.

5. Performance Benchmarks and Experimental Validation

LASEM models have demonstrated superior or competitive accuracy and significant computational speed advantages:

• SPAs: RMSE improved from 6 N (physics-only) to sub-2 N (with AL correction), and real-time evaluation achieved at sub-ms latency (Campbell et al., 1 Apr 2025).
• CDCRs: In static settings, tip position error is $<0.5$ mm ($\approx$0.2%) compared to Cosserat-rod models; for dynamic simulation, Galerkin-reduced LASEM achieves an average 62.3% speedup over state-of-the-art reduced Cosserat-rod solvers, with evaluations (<100 μs analytic, 100–200 μs inverse update) that reliably support high-rate control (Wu et al., 4 Sep 2025, Yang et al., 15 Dec 2025).
• Manipulators: When trained on real data, models yield root-mean-square errors as low as 1.42 W (UR3e) in power prediction, and $R^2$ values up to 0.975 on test sets, generalizing well to unseen trajectories (Heredia et al., 8 Aug 2025).
• Thermostatic Loads: Bilinear population models match empirically measured aggregate power to within statistical noise, and empirical computational cost is two orders of magnitude below that of large-scale Monte Carlo simulation at the same population size (Totu et al., 2014).

6. Limitations, Assumptions, and Future Directions

LASEM’s utility depends on appropriate construction of actuation-space energetic or statistical models:

• Neglected phenomena: In the continuum robot context, friction is typically neglected; for high-fidelity scenarios, this assumption may need augmentation (Wu et al., 4 Sep 2025).
• Empirical correction need: Purely analytic models can retain non-negligible prediction errors without data-driven corrections, particularly in soft actuator systems; hybrid approaches (employing active learning or regression) are thus recommended for high-accuracy demands (Campbell et al., 1 Apr 2025, Heredia et al., 8 Aug 2025).
• Discretization artifacts: When physical robots are assembled from rigid segments rather than truly continuous media, numerical optimization is required, and care must be taken to ensure basis function richness for convergence (Wu et al., 4 Sep 2025).
• Stochastic actuation variability: Aggregated population models assume sufficiently large and homogeneous device samples; for small or highly heterogeneous populations, additional macroscopic stochastic effects may arise (Totu et al., 2014).

A plausible implication is that future research will expand LASEM to even more general multiactuator, hybrid, and variable-structure systems, potentially integrating model-based and data-driven elements at multiple layers, and exploring fully uncertainty-aware prediction for robust real-time control.

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Key References:

• "Active Learning Design: Modeling Force Output for Axisymmetric Soft Pneumatic Actuators" (Campbell et al., 1 Apr 2025)
• "Lightweight Kinematic and Static Modeling of Cable-Driven Continuum Robots via Actuation-Space Energy Formulation" (Wu et al., 4 Sep 2025)
• "Lightweight Dynamic Modeling of Cable-Driven Continuum Robots Based on Actuation-Space Energy Formulation" (Yang et al., 15 Dec 2025)
• "EcBot: Data-Driven Energy Consumption Open-Source MATLAB Library for Manipulators" (Heredia et al., 8 Aug 2025)
• "Modeling Populations of Thermostatic Loads with Switching Rate Actuation" (Totu et al., 2014)