Lightweight Actuation Space Energy Modeling

• Lightweight Actuation Space Energy Modeling (LASEM) is a framework that directly models actuation variables like cable displacements and transmission ratios to optimize energy efficiency.
• It integrates analytical methods and convex optimization to provide real-time control, robust design, and multiobjective co-design across diverse robotic applications.
• LASEM enables uncertainty quantification and hybrid system integration, driving innovation in continuum robots, series elastic actuators, and smart actuator designs.

Lightweight Actuation Space Energy Modeling (LASEM) refers to a family of analytical and computational frameworks that model, optimize, and predict the energetic and mechanical behavior of actuation systems—particularly those designed to minimize weight and maximize energy efficiency. LASEM approaches formulate system models directly in actuation space, focusing on variables such as cable length changes, actuator forces, transmission ratios, or elastic element profiles. These models enable rigorous energy-based design and control across a range of robotic and mechatronic domains, from continuum robots to wearable devices and industrial manipulators.

1. Actuation-Space Energy Formulation in Cable-Driven Continuum Robots

The LASEM framework for cable-driven continuum robots formulates the system’s potential energy directly in the actuation space, yielding analytical models derived from geometrically nonlinear beam and rod theories via Hamilton’s principle (Wu et al., 4 Sep 2025). The total potential energy incorporates both elastic (strain) energy and actuation potential energy (negative work of cable forces or displacements), producing closed-form solutions for the static and kinematic configuration.

Key equations include: $$\Pi = \int_0^L \frac{1}{2}EI \left( \frac{d\theta}{ds} \right)^2 ds - (F_1\Delta l_1 + F_2\Delta l_2)$$

$\Delta l = \frac{W\theta(L)}{2}$

where $EI$ is bending stiffness, $\theta(s)$ is backbone angle along arc length $s$, $F_{i}$ are cable forces, and $\Delta l_{i}$ are cable displacements. The variational principle yields equilibrium equations, e.g., $EI \frac{d^2\theta}{ds^2} = 0$ (for uniform bending). Solutions accept both force and displacement inputs, thereby unifying static and kinematic analyses.

This approach accommodates nonuniform geometries, arbitrary cable routings, distributed loading, and axial extensibility under the assumption of negligible friction and without explicit cable–backbone contact modeling. Computational efficiency is ensured by directly deriving analytical forms for the backbone shape, which are suitable for real-time control and planning. Discretized versions are implemented via polynomial approximations and numerical optimization, where cable potential energy is naturally incorporated as actuation input constraints.

2. Convex Optimization and Multiobjective Design of Series Elastic Actuators

LASEM is applied to series elastic actuators (SEAs) by embedding elastic element design within a convex optimization framework (Bolívar et al., 2018). Here, the torque–elongation profile $f(\delta)$ for springs is determined “non-parametrically” via optimization over the motor’s discrete position trajectory vector, subject to quadratic energy cost functions and affine constraints guaranteeing conservative spring operation and actuator limitations.

Discretized motor energy is expressed as: $E_m \approx q_m^T Q_e q_m + A_e q_m + c_e$ where $q_m$ is the motor position vector, and the matrices $Q_e, A_e, c_e$ encode system and trajectory parameters. Monotonicity and physical limits are imposed via linear or equality constraints derived from trajectory differences. Peak power is treated with a convex surrogate by neglecting inertial contributions: $p_m^{cvx}(i) = q_m^T G_i^{cvx} q_m + H_i q_m$ Multiobjective optimization is enabled by a weighted sum of energy consumption and peak power: $$\min \; \theta \cdot \gamma_2 (q_m^T Q_e q_m + A_e q_m + c_e) + (1 - \theta) [\max\{p_m^{cvx}\} + \gamma_1\|D_2 q_m\|_\infty]$$ with trade-off parameters $\theta, \gamma_1, \gamma_2$.

Case studies demonstrate that nonlinear SEA designs significantly reduce energy and peak power, extend achievable task ranges, and enable real-time iterative design and control. The framework is also robust to actuator constraints such as maximum torque, motor speed, and spring deflection limits.

3. Robust Design under Uncertainty for Energy Efficient Actuators

Extensions of LASEM address uncertainty in actuator design by incorporating uncertainty sets into a convex quadratic optimization formulation (Bolívar et al., 2018). Here, compliance $\alpha = 1/k$ is the decision variable, and the motor energy consumption is modeled as: $E_m = a\alpha^2 + b\alpha + c$ Uncertainties in parameters such as load kinematics, transmission efficiency, and manufacturing tolerances are modeled via interval sets (e.g., $$U_{q_l} = \{q_l \in \mathbb{R}^n : \bar{q}_l - \epsilon_{q_l} \leq q_l \leq \bar{q}_l + \epsilon_{q_l}\}$$). Constraints (on maximum torque, velocity, spring elongation) are robustified by requiring satisfaction for all values within the uncertainty sets, yielding conservative bounds in the optimization.

An analytical solution exists for the unconstrained case ($\alpha^* = -b/(2a)$), and the robust formulation ensures feasibility under uncertain operation with minimal loss of energy efficiency. Simulation on powered prosthetic ankles demonstrates that small increases in stiffness, resulting from robust optimization, guarantee actuator performance across uncertainties with only minor compromises in energy savings.

4. Multiscale and Multiphysics Optimization in Lightweight Porous Actuators

LASEM concepts are embedded in topology optimization frameworks for laser-activated porous actuators (Ali et al., 23 May 2024). A concurrent multiscale optimization treats macrostructure ($P_M$) and microstructure ($p_m$) design variables to maximize actuation displacement subject to thermal–mechanical equilibrium and volume constraints: $$\max\; u_{\text{out}}(P_M, p_m) \quad \text{s.t.} \quad K(P_M, p_m) U = F_{\text{th}}, \quad \int_\Omega p\,d\Omega \leq v$$ Laser-induced heating is modeled with both Gaussian intensity distributions and a complete set of heat dissipation mechanisms (conduction, convection, radiation). Homogenization transfers microscale thermal and elastic properties to the macroscale model. Sensitivity analysis via the adjoint method enables gradient-based optimization.

Numerical studies demonstrate that porous microstructure designs up to 75% weight reduction outperform solid actuators in displacement and efficiency, with concurrent optimization outperforming sequential macro-then-micro approaches. The incorporation of radiation as a major heat loss mechanism represents a critical advancement over previous models.

5. Co-Design and Actuation-Space Optimization in Robotics with Parallel Transmissions

LASEM extends to robotic co-design in the context of parallel belt transmissions, where actuation-space modeling leads to improved dynamic payload capacity and energy performance (Kumar et al., 1 Jul 2025). The robot’s dynamic model is transformed via a coupling matrix $G$ to actuation space: $$q_u = Gq, \quad \dot{q}_u = G\dot{q}, \quad \ddot{q}_u = G\ddot{q} + g_u$$ with actuation-space dynamics: $$H_u\ddot{q}_u + C_u = \tau_u, \quad \text{where } H_u = G^{-T}HG^{-1}, \tau_u = G^{-T}\tau$$ A bi-level co-design framework adjusts transmission ratios (elements of $G$) in the outer loop, while actuation-space trajectory optimization occurs in the inner loop. This approach yields better utilization of actuator limits and, in empirical studies, doubles the payload capability compared to serial-only modeling.

Energy modeling benefits include reduced consumption, improved distribution of torque requirements, and enhanced battery life prediction owing to alignment of dynamic requirements with true actuation capacity.

6. Modeling and Design in Soft Pneumatic Actuators via Energy Minimization and Active Learning

Within LASEM, soft pneumatic actuators are modeled by solving energy minimization equations that capture the force–pressure–height relationship for axisymmetric membranes under internal pressure and external load (Campbell et al., 1 Apr 2025). The elastic strain energy for axisymmetric membranes is computed via the Gent hyperelastic model, with equilibrium equations for principal stretches and tangent angle derived from the first variation of the energy functional.

For heterogeneous/ring-reinforced designs, the model partitions the domain and solves the equilibrium equations piecewise, matching boundary conditions at material interfaces. These physics-based models are compared with neural network–based learned models employing a Ring Encoder, which interpolate between empirical data and are differentiable for design optimization. The learned models outperform theory-based and naive regression models, achieving RMSE as low as 4% of maximum force.

Optimized membrane designs for lifting tasks demonstrated superior accuracy and trajectory performance, and the framework supports intelligent lifts in single-pressure input systems by enabling design-based mapping of membrane geometry to force output.

7. Hybrid and Multimodal Approaches for Wearable and Lightweight Robots

LASEM is influential in hybrid actuation systems, particularly multimodal hydrostatic actuators for wearable robots (Denis et al., 2022). Dynamic reconfiguration using hydraulic valves enables on-the-fly switching between high-force/geared and high-speed/low-geared power sources, reducing actuator weight and improving efficiency.

The design includes energy accumulators for supplying continuous preload force, which allows sizing motors for lower continuous requirements, and hydraulic locking mechanisms that cut energy use during hold states. Formulaic approaches optimize gear ratios and estimate mass reductions via scaling laws: $J_{\text{reflected}} = J_{\text{motor}} N^2$

$$\tau_{\text{motor}} = \frac{\tau_{\text{joint}}}{\eta_{\text{trans}} N}$$

Mass savings are quantifiable and designable under LASEM, enabling sizing for each operating point rather than the worst case, and improvement in battery life for wearable devices working in variable-cycle conditions.

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LASEM frameworks connect mechanics, optimization, and control in actuation systems, directly leveraging actuation-space variables to achieve energy-efficient, lightweight, and dynamically robust designs. Applications span continuum robotics, industrial manipulators, soft actuators, multimodal hydrostatic systems, and more. The approach enables real-time control, analytical solutions, and robust design under uncertainty, advancing the efficiency and capability of next-generation robotic and mechatronic systems.