Scaling Mechanical Output over Lifetime (SMOL)

• SMOL is a framework that schedules and constrains robotic actuation to maximize useful mechanical output over extended cycles rather than short-term peak force.
• It integrates methods from SMA soft robots, torque scheduling in locomotion curricula, and fatigue analysis in elastic manipulators to address degradation and performance trade-offs.
• The approach emphasizes designing controllers and structures that account for internal damage and wear, guiding choices in temperature limits, torque scaling, and material thickness for longevity.

Searching arXiv for the provided SMOL-related papers and closely related work to ground the article in the current record. Scaling Mechanical Output over Lifetime (SMOL) denotes a class of robotics formulations in which mechanical capability is scheduled, constrained, or dimensioned with respect to performance over extended use rather than instantaneous peak output. In the current arXiv record, the term appears explicitly as a developmental curriculum for locomotion discovery in which actuator torque is scaled over the course of training (Templier et al., 15 Sep 2025). Closely related work uses the same underlying objective without always using the acronym: electrothermal shape memory alloy (SMA) actuation is controlled to maximize consistent force over many cycles (Anderson et al., 2024), and elastic-link manipulators are dimensioned using fatigue-based lifetime estimation so that throughput, vibration, and structural longevity can be traded off quantitatively (Zauner et al., 27 Oct 2025). A still earlier hardware-reliability precursor asked about the dependency of component lifetime on dynamic voltage scaling, indicating a conceptually similar concern with lifetime-aware operating policies, although that arXiv entry exposes only a title and one-line description rather than a recoverable full text (Jaberi, 2012).

1. Core formulation and unifying objective

Across the available literature, SMOL is not a single algorithm but a unifying optimization perspective: select an operating schedule, controller, or structure that preserves useful mechanical output across a long horizon of cycles, task repetitions, or training phases. The common contrast is between maximizing immediate force or torque and maximizing sustained output under degradation, fatigue, or embodiment change.

Domain	Primary decision variable	Lifetime-oriented quantity
SMA soft limb	Maximum internal temperature $T_{\text{SET}}$ with supervisory clipping	Asymptotic blocked force $F_{\max,\infty}$ and consistent force over cycles
QD locomotion training	Torque scaling factor $\alpha(k)$ over 100 phases	Final max fitness and coverage at standard torque $\alpha=1$
Elastic-link manipulator	Link wall thicknesses and resulting damage per task $D$	$t_{\text{life}} = t_{\text{task}}/D$ and $N_{\text{life}} = 1/D$

In the SMA setting, the paper does not explicitly compute “lifetime mechanical output,” but it makes the implied SMOL quantity explicit as

$\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$

and further notes that a practical long-life choice should maximize $F_{\max,\infty}(T_{\text{SET}})$ while keeping the decay rate small (Anderson et al., 2024). In the elastic-manipulator setting, the analogous lifetime quantity is the number of repeated tasks before fatigue initiation, obtained from the damage per task, $N_{\text{life}} = 1/D_{\max}$, which directly converts mechanical repetition into lifetime-limited output (Zauner et al., 27 Oct 2025). In the locomotion-curriculum setting, lifetime is the learning horizon itself: the body is deliberately changed over 100 phases so that the final 10 phases are always evaluated under the same standard embodiment, $F_{\max,\infty}$0, making the curriculum a mechanism for discovering behaviours that survive embodiment change (Templier et al., 15 Sep 2025).

This unification suggests that SMOL is best understood as a control-and-design doctrine in which the relevant state is not only pose, force, or reward, but also remaining actuator or structural viability.

2. Antecedents and conceptual lineage

The earliest item in the present corpus is “An Open Question about Dependency of Life Time of Hardware Components and Dynamic Voltage Scaling” (Jaberi, 2012). The arXiv entry ([1203.3909](/papers/1203.3909))v4 has no PDF or source associated with it, and only the single-line description is accessible. As a result, no formulas, experiments, or direct technical claims can be attributed to that paper beyond its title-level concern.

Even so, the title establishes an antecedent question that is structurally aligned with later SMOL work: how an operating policy that improves short-term efficiency may alter lifetime through accumulated stress. The supplied interpretation of that entry associates dynamic voltage scaling with histories $F_{\max,\infty}$1, $F_{\max,\infty}$2, and $F_{\max,\infty}$3, and with a trade-off between instantaneous performance and lifetime-integrated work. Because the full text is unavailable, this should be treated as interpretive rather than documentary (Jaberi, 2012).

The later robotics papers make that abstract concern concrete in two distinct ways. One line of work constrains an internal physical state that mediates both output and fatigue, namely SMA temperature, and enforces a long-life operating region online (Anderson et al., 2024). Another changes actuator capability over the course of learning, not to protect hardware from wear, but to exploit a developmental sequence of embodiments that improves the final controller under standard conditions (Templier et al., 15 Sep 2025). A third computes stress histories, rainflow count matrices, and Miner damage for flexible manipulator links so that structural life becomes an explicit post-design selection criterion (Zauner et al., 27 Oct 2025).

Taken together, these works show that the core SMOL question is not domain-specific. It recurs whenever output is coupled to an internal damage variable, a fatigue envelope, or a schedule of embodiment change.

3. Temperature-limited SMOL in SMA soft robots

In “Maximizing Consistent Force Output for Shape Memory Alloy Artificial Muscles in Soft Robots,” the central SMOL object is a temperature limit that maximizes consistent force over many cycles (Anderson et al., 2024). The paper treats SMA temperature as the key internal state that mediates both force and fatigue. Temperature dynamics are modeled as a first-order linear system driven by PWM input $F_{\max,\infty}$4,

$F_{\max,\infty}$5

with identified parameters for the specific coiled Nitinol actuator at $F_{\max,\infty}$6 s: $F_{\max,\infty}$7, $F_{\max,\infty}$8, and $F_{\max,\infty}$9. Rather than modeling phase fraction or strain explicitly, the paper uses blocked force as an empirical proxy for mechanical output under fixed geometry.

The mechanical system is a cast silicone limb with centrally embedded coiled SMA wires, a modular bracket for swap-able SMA bundles, and a thermocouple attached directly to the SMA bundle via thermally conductive, electrically insulating epoxy. The blocked-force test holds limb geometry essentially fixed so that the measured endpoint force is effectively blocked bending force. Two cycling profiles are used. C1 is temperature-based: nominal heating proceeds until the measured temperature is within $\alpha(k)$0 of $\alpha(k)$1, a 20 s hold is executed under supervisory temperature limitation, then the actuator cools until $\alpha(k)$2. C2 is time-based: nominal heating is applied for 45 s and cooling for 65 s, with the supervisory layer clipping the command if the predicted temperature would exceed the setpoint. Each trial uses $\alpha(k)$3 cycles.

For each cycle $\alpha(k)$4, the paper extracts a maximum blocked force $\alpha(k)$5 and fits either a single-exponential or double-exponential decay model,

$\alpha(k)$6

or

$\alpha(k)$7

choosing between them by RMSE. The asymptotic term $\alpha(k)$8 is interpreted as the predicted long-life steady blocked force,

$\alpha(k)$9

Plotting $\alpha=1$0 against temperature shows a plateau of approximately $\alpha=1$1–$\alpha=1$2 N from about $\alpha=1$3 to $\alpha=1$4. Above approximately $\alpha=1$5, long-life predictions diverge between profiles and become noisy, with no clear increase in sustained force. The paper therefore defines a conservative long-life limit

$\alpha=1$6

described as the lowest temperature inflection where $\alpha=1$7 stops increasing and begins to become profile-dependent while still giving high asymptotic force across both profiles.

The online controller turns that empirical characterization into a real-time policy. A nominal input generator proposes $\alpha=1$8, while a supervisory controller computes the maximum control input compatible with the temperature bound: $\alpha=1$9 with $D$0 in experiments. The corresponding safe set is

$D$1

and the control law is designed so that the next temperature remains inside that set. In the paper’s terminology, this is invariance-based supervisory control.

The validation experiment makes the SMOL logic explicit. Two SMA bundles are compared in a free-motion lifting task with a 50 g tip load: one had previously been fatigued under C1 at $D$2, and the other under C1 at $D$3. Both are then operated at $D$4. The bundle previously cycled at $D$5 shows significantly lower displacement, approximately half the displacement of the $D$6-fatigued limb, while the $D$7 limb maintains large, consistent displacements over hundreds of cycles. This directly refutes the common intuition that higher temperature is mechanically preferable whenever it yields larger short-term actuation: above the long-life limit, additional heating does not materially increase sustainable force, but it does degrade later output.

The paper’s limitations are also SMOL-relevant. The experiments are blocked-force tests, not force–displacement work measurements; there is no explicit phase-fraction or microstructure model; only two profiles are explored; variability in hand-fabricated commercial SMAs affects threshold precision; and the controller does not estimate degradation online. Those limitations imply that the present formulation identifies a high-value invariant set in temperature space, but not yet a full multidimensional health state over temperature, strain, and internal stress.

4. Developmental SMOL as torque scheduling in quality-diversity search

“Time to Play: Simulating Early-Life Animal Dynamics Enhances Robotics Locomotion Discovery” introduces Scaling Mechanical Output over Lifetime as a developmental curriculum for robot training (Templier et al., 15 Sep 2025). Here SMOL does not model physical fatigue. Instead, it changes actuator strength over the course of learning so that the agent experiences a sequence of embodiments before being evaluated under standard conditions.

The mechanism is uniform actuator-torque scaling. If the original actuator gear is $D$8, the modified gear is

$D$9

which is equivalent, under the paper’s ideal-torque actuator model, to scaling the maximum torque: $t_{\text{life}} = t_{\text{task}}/D$0 Training is divided into 100 phases indexed by $t_{\text{life}} = t_{\text{task}}/D$1, and all schedules satisfy

$t_{\text{life}} = t_{\text{task}}/D$2

so the final 10 phases are always evaluated under identical physics.

The baseline SMOL schedule is a monotonic decline from a strong early body to the standard one: $t_{\text{life}} = t_{\text{task}}/D$3 and $t_{\text{life}} = t_{\text{task}}/D$4 thereafter. The reverse schedule begins at $t_{\text{life}} = t_{\text{task}}/D$5 and rises linearly to $t_{\text{life}} = t_{\text{task}}/D$6. SMOL-Human is bell-shaped, motivated by empirical human power-per-kilogram curves, and is described qualitatively as $t_{\text{life}} = t_{\text{task}}/D$7, with the last 10 phases again fixed at $t_{\text{life}} = t_{\text{task}}/D$8. Additional baselines are Vanilla MAP-Elites with $t_{\text{life}} = t_{\text{task}}/D$9, a Random schedule with $N_{\text{life}} = 1/D$0 before phase 90, and Extinction baselines that keep $N_{\text{life}} = 1/D$1 but randomly delete archive entries with probability $N_{\text{life}} = 1/D$2.

The curriculum is integrated into MAP-Elites by archive reevaluation at every phase boundary. After $N_{\text{life}} = 1/D$3 is updated to $N_{\text{life}} = 1/D$4, every solution $N_{\text{life}} = 1/D$5 in the archive is re-simulated under the new physics, its fitness and descriptor are recomputed, and the reevaluated solutions are inserted into a new archive $N_{\text{life}} = 1/D$6 according to standard MAP-Elites replacement rules. The new archive seeds the next phase. The paper describes this as a form of developmental memory: policies discovered under earlier embodiments are not discarded but transferred and adapted under the new torque limits.

Experiments use Brax v2 with the spring backend on HalfCheetah, Walker2D, and Ant Omni. HalfCheetah and Walker2D maximize forward velocity over 1000 environment steps; Ant Omni minimizes energy use and uses final $N_{\text{life}} = 1/D$7 position after 100 steps as the descriptor. Controllers are feedforward neural networks with tanh activations, with hidden layers $N_{\text{life}} = 1/D$8 for HalfCheetah and Walker2D and $N_{\text{life}} = 1/D$9 for Ant Omni. Descriptor spaces are discretized into 1024 cells using Centroidal Voronoi Tessellation. Mutation uses iso+line noise with $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$0 and $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$1. Total evaluations per run are $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$2, with 7–10 seeds per condition. The principal metrics are max fitness,

$\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$3

and coverage,

$\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$4

The main finding is that all schedule-based methods match or outperform static physics on final metrics, and that the monotonic high-to-normal SMOL schedule is consistently best or statistically tied with the best. For Ant Omni, median final coverage is $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$5 for SMOL versus $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$6 for Vanilla, with $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$7. For HalfCheetah, median final max fitness is $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$8 for SMOL versus $\sum_{v=1}^{V} F_{\max}(v; T_{\text{SET}})$9 for Vanilla, with $F_{\max,\infty}(T_{\text{SET}})$0. For Walker2D, median final max fitness is $F_{\max,\infty}(T_{\text{SET}})$1 for SMOL versus $F_{\max,\infty}(T_{\text{SET}})$2 for Vanilla, with $F_{\max,\infty}(T_{\text{SET}})$3. SMOL also outperforms the Random schedule and, in Walker2D and HalfCheetah, exceeds SMOL-Human at conventional significance thresholds.

The paper attributes the gains to two linked mechanisms. First, high early torque broadens the set of physically reachable behaviours, making dynamic gaits easier to discover and populating the archive with stepping stones. Second, the changing torque schedule alters selective pressure over time. The observed drops in coverage and fitness at schedule transitions show that archive reevaluation is not benign; many elites become suboptimal when torque changes. The extinction ablation demonstrates that archive disruption alone can help exploration, especially at $F_{\max,\infty}(T_{\text{SET}})$4, but does not match the full SMOL effect. The random-schedule ablation shows that unstructured embodiment change can help in some cases but is unstable and generally inferior to a structured developmental schedule. A central misconception is therefore ruled out: SMOL is not equivalent to arbitrary domain randomization over body parameters.

The paper also sharply delimits its own scope. Only actuator torque changes; morphology and mass remain fixed. All results are in simulation. Only locomotion tasks are studied. SMOL-Human is biologically motivated but underperforms the simpler monotonic schedule, suggesting that schedule realism and schedule usefulness need not coincide when the embodiment change is restricted to uniform torque scaling.

5. Fatigue-informed SMOL in elastic-link manipulators

“Optimal Dimensioning of Elastic-Link Manipulators regarding Lifetime Estimation” provides a structural and fatigue-analytic counterpart to SMOL (Zauner et al., 27 Oct 2025). The paper does not present a curriculum, but it does provide a direct mapping from design and task execution to lifetime-limited mechanical output. Its basic lifetime estimate is

$F_{\max,\infty}(T_{\text{SET}})$5

where $F_{\max,\infty}(T_{\text{SET}})$6 is the duration of one representative repeated task and $F_{\max,\infty}(T_{\text{SET}})$7 is the fatigue damage increment per task at a critical point. This yields a direct cycle-life quantity,

$F_{\max,\infty}(T_{\text{SET}})$8

once the most damaging cutting plane has been identified.

The manipulator is a 3-DOF articulated serial robot with gearbox elasticities and link elasticities. Flexible links are modeled as Euler–Bernoulli beams with small deflections, isotropic linear elastic material, and square hollow cross-sections. The reference cross-section has edge length $F_{\max,\infty}(T_{\text{SET}})$9, and the link wall thicknesses $N_{\text{life}} = 1/D_{\max}$0 and $N_{\text{life}} = 1/D_{\max}$1 are the design variables. The equations of motion are written in flexible multibody form,

$N_{\text{life}} = 1/D_{\max}$2

with elastic restoring force

$N_{\text{life}} = 1/D_{\max}$3

Lifetime estimation proceeds by transforming multiaxial stress histories into a scalar fatigue-driving signal. At a critical point, the paper assumes plane stress and introduces a cutting plane rotated by angle $N_{\text{life}} = 1/D_{\max}$4 in the $N_{\text{life}} = 1/D_{\max}$5-plane. The shear stress on that plane is

$N_{\text{life}} = 1/D_{\max}$6

The equivalent uniaxial stress is then defined by the Tresca hypothesis as

$N_{\text{life}} = 1/D_{\max}$7

For each cutting angle, the resulting stress history is reduced to extrema and processed by rainflow counting according to ASTM E1049, producing counts $N_{\text{life}} = 1/D_{\max}$8 over mean-stress and amplitude-stress cells. Fatigue resistance is obtained from a Haigh diagram and a synthetic log–log Wöhler line between $N_{\text{life}} = 1/D_{\max}$9 and $F_{\max,\infty}$00. Damage is accumulated linearly by Palmgren–Miner: $F_{\max,\infty}$01

The design study evaluates a 6-by-6 grid of candidate geometries with $F_{\max,\infty}$02, giving 36 configurations. The primary optimization is bi-objective: minimize a mass-related objective and an end-effector vibration objective, with lifetime used afterward as a selection criterion on the Pareto set. Qualitatively, decreasing thickness reduces stiffness and increases end-effector oscillation. However, increasing thickness indefinitely is not uniformly beneficial, because heavier links can amplify gearbox-compliance effects and worsen vibration. This makes the problem intrinsically coupled: stiffness, mass, vibration, and fatigue cannot be separated.

The reported Pareto front includes configurations $F_{\max,\infty}$03. Configuration 1 achieves approximately $F_{\max,\infty}$04 mass reduction but exhibits high vibration at $F_{\max,\infty}$05 and a finite lifetime of approximately $F_{\max,\infty}$06. Configurations 2–6 show smaller oscillations, in the range $F_{\max,\infty}$07–$F_{\max,\infty}$08, and are plotted at $F_{\max,\infty}$09, interpreted in the paper as effectively infinite lifetime in the considered range because the stress amplitudes do not exceed fatigue strength. Other selected configurations show sharply shorter lives: configuration 7 at approximately $F_{\max,\infty}$10, configuration 13 at $F_{\max,\infty}$11, configuration 19 at $F_{\max,\infty}$12, configuration 25 at $F_{\max,\infty}$13, and configuration 31 at $F_{\max,\infty}$14. The key geometric insight is that thinning the first link is especially detrimental because the proximal link carries more load; equivalent mass reduction in a distal link is less damaging.

From a SMOL standpoint, the paper converts structural design into an output-over-life problem. If the task is repetitive pick-and-place, then the total number of parts moved before fatigue initiation is proportional to $F_{\max,\infty}$15. The paper itself fixes the task trajectory and optimizes geometry, but its methodology directly supports a broader formulation in which trajectory aggressiveness and geometry are co-optimized under a lifetime constraint.

6. Common themes, misconceptions, and unresolved questions

The three mature SMOL formulations in the present literature share a common structure: a controllable or designable variable changes instantaneous capability, but also alters the regime in which future output remains achievable. In the SMA paper, the internal state is temperature and the health-preserving object is the invariant set $F_{\max,\infty}$16 (Anderson et al., 2024). In the locomotion paper, the state variable is not damage but phase-indexed embodiment, and the relevant mechanism is archive transfer across a scheduled torque history $F_{\max,\infty}$17 (Templier et al., 15 Sep 2025). In the manipulator paper, the internal quantity is accumulated fatigue damage computed from stress histories and cycle counting (Zauner et al., 27 Oct 2025).

Several misconceptions are explicitly contradicted by these results. First, maximum instantaneous actuation does not imply maximum lifetime-integrated output. The SMA data show that increasing maximum temperature above approximately $F_{\max,\infty}$18 yields marginal short-term gains but worsens sustainable performance, and prior cycling at $F_{\max,\infty}$19 leaves the limb with approximately half the displacement of a $F_{\max,\infty}$20-limited limb in later operation (Anderson et al., 2024). Second, structured embodiment change is not interchangeable with random embodiment change. In locomotion discovery, the Random schedule is unstable and generally inferior to the monotonic SMOL schedule, while extinction-style archive disruption alone does not account for the gains (Templier et al., 15 Sep 2025). Third, lightweight design is not synonymous with efficient lifetime use. In elastic-link manipulators, aggressive wall-thickness reduction can move a design onto a poor fatigue regime even if it improves nominal mass metrics (Zauner et al., 27 Oct 2025).

The unresolved problems are likewise domain-specific but conceptually aligned. For SMA systems, the open issues include the absence of direct work-per-cycle measurements, limited loading profiles, no explicit strain or microstructure model, and no online damage-state observer (Anderson et al., 2024). For developmental torque scaling, the open issues include richer embodiment change beyond uniform torque, real-robot implementation, schedule tuning, and extension beyond locomotion (Templier et al., 15 Sep 2025). For elastic manipulators, the current formulation estimates lifetime for a representative repeated task but does not yet jointly optimize trajectory duration, structural geometry, and damage within a single closed design loop (Zauner et al., 27 Oct 2025). The 2012 dynamic-voltage-scaling precursor reinforces that this broader family of questions extends beyond robotics proper: once operating policy affects both output and degradation, SMOL-like reasoning becomes unavoidable, even if the precise lifetime model remains unsettled (Jaberi, 2012).

In that sense, SMOL is best regarded not as a narrow recipe but as a general research program: determine the internal variable that couples capability to degradation or developmental adaptation, characterize how output evolves across repeated use or scheduled embodiment change, and then choose operating limits or schedules that maximize useful mechanical performance over the relevant lifetime horizon.