Optimizing energy consumption for legged robot by adapting equilibrium position and stiffness of a parallel torsion spring

Artem Erkhov; Danil Belov; Dzmitry Tsetserukou; Elisaveta Pestova; Farit Khabibullin; Ilya Osokin; Pavel Osinenko; Sergei Satsevich

REVIEW 4 major objections 7 minor 17 references

Optimizing energy consumption for legged robot by adapting equilibrium position and stiffness of a parallel torsion spring

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A torsion spring with closed-form optimal stiffness and equilibrium position can reduce simulated energy consumption of a cyclic legged-robot task to as little as 0.15% of baseline.

desk verdict A clean closed-form least-squares fit for parallel spring parameters, but the headline energy savings are in-sample residuals of a torque-squared proxy, not measured total energy. read the letter →

arxiv 2411.18295 v1 pith:M7AHXHRF submitted 2024-11-27 cs.RO

Danil Belov , Artem Erkhov , Farit Khabibullin , Elisaveta Pestova , Sergei Satsevich , Ilya Osokin , Pavel Osinenko , Dzmitry Tsetserukou This is my paper · ORCID

classification cs.RO

keywords adaptivecompliancetorsionspringenergyefficiencyleggedrobotsparallelelasticactuatorclosed-formoptimizationGazebosimulationcyclicmotion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that a single torsion spring mounted in parallel with the knee actuator of a legged robot can be tuned, in one closed-form step, to compensate most of the torque the motor would otherwise produce during a cyclic motion. The tuning rule minimizes the sum of squared actuator torques over a recorded trajectory, yielding explicit formulas for the spring's stiffness and equilibrium position. In Gazebo simulations of a three-degree-of-freedom leg performing vertical oscillations, the spring-equipped system consumes as little as 0.15% of the baseline energy, with the reduction holding across changes in mass, frequency, amplitude, and starting height. The point of the work is that a mechanically simple adjustment could extend battery life and reduce servo wear without iterative learning or online search.

What carries the argument

The central object is the quadratic energy functional over one motion cycle, $E = K\sum_{i=1}^T (\tau_i - \mu(\alpha_i - \alpha_0))^2 \Delta t$, where $\tau_i$ is the actuator torque, $\alpha_i$ the joint angle, $\mu$ the spring stiffness, and $\alpha_0$ the spring equilibrium position. Treating the spring torque as a linear regressor for the actuator torque turns parameter optimization into a linear least-squares problem, and Equation 4 gives the closed-form normal-equation solution for $\mu^*$ and $\alpha_0^*$. This is what carries the argument: no iterative search, no learning, just a direct computation from a single recorded trajectory.

What would settle it

Run the same cyclic motion in the simulator (or on the physical stand) with the spring installed using the baseline-optimal $\mu^*$ and $\alpha_0^*$, then record the actual actuator torque after the spring is in place; if the measured energy ratio differs substantially from the predicted $E_a/E_0$, the one-step closed-form prediction is invalid. Alternatively, allow the controller to re-optimize the trajectory with the spring present: if this yields markedly lower energy than the paper's predicted ratio, the two-step procedure is suboptimal.

Watch

Extended reading notes

Core claim

For a repeated cyclic task, the energy-optimal linear torsion spring is the one whose torque, $\mu(\alpha - \alpha_0)$, is the least-squares fit to the actuator torque profile over the cycle. The paper derives the closed-form solution for $\mu$ and $\alpha_0$ by differentiating the energy functional $E = K\sum_{i=1}^T (\tau_i - \mu(\alpha_i - \alpha_0))^2 \Delta t$ and solving the resulting normal equations. Applying these parameters in simulation cancels nearly all of the knee motor's torque, yielding energy ratios $E_a/E_0$ from 0.15% to 3.9% across the tested conditions. The authors present this as evidence that an adaptive torsion spring with adjustable equilibrium position can considerably reduce energy consumption in cyclic legged locomotion.

Load-bearing premise

The spring parameters are computed from a trajectory recorded without the spring, and the paper assumes the actuator torque at each joint angle stays the same once the spring is added, even though the spring changes the load dynamics.

Editorial extensions

If this is right

In simulation, the spring reduces energy consumption to between 0.15% and 3.9% of baseline across varying mass, frequency, amplitude, and starting height, implying near-total torque cancellation in the tested cyclic task.
The optimal parameters depend on the task conditions (stiffness ranges from 6.07 to 17.1, equilibrium position from -1.4 to -2.84), so the adjustable equilibrium position is essential rather than decorative.
Lower actuator torque implies less heat generation and less mechanical wear on the servos, which the paper cites as a direct practical benefit.
Because the parameters come from a closed-form solution, the approach is computationally light and could run in real time or be set once per task without training data.
The proposed mechanism, a torsion spring with a worm-gear-driven preload adjustment, extends the same idea to hardware, though the current evidence is entirely from the Gazebo simulator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The closed-form solution is mathematically a least-squares projection of the torque profile onto the affine line spanned by the joint angle, so its 'optimality' is only with respect to the squared-torque cost and the trajectory recorded without the spring; the paper does not prove that the spring leaves the torque-angle relationship unchanged.
A natural test the authors do not run: replan the leg trajectory with the spring in place and re-optimize. Their one-shot baseline-to-spring procedure may be conservative; a joint optimization could yield even lower energy or expose that the spring's added inertia shifts the optimal parameters.
The same derivation applies to any cyclic joint torque, not just the knee in this stand setup, so the formula could transfer to other joints or gaits; validating it on a walking gait with multiple coupled joints would be the next step.
In a physical system, friction and the servo that adjusts the preload consume energy themselves; the reported ratios are for the ideal spring model, and a hardware test would need to account for these overheads before claiming net savings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the math is right, the mechanism is plausible, but Table I does not establish overall energy consumption reduction. The paper fits spring parameters to a baseline trajectory, then measures the torque-squared metric on that same trajectory. The 0.15%–3.9% ratios are essentially the residual variance of a linear fit; they tell you how well a linear spring approximates the baseline torque curve, not how much battery energy you save.

What is genuinely useful: the closed-form solution (Eq. 4) is the correct least-squares optimum for the stated proxy, and the torsion-spring-with-worm-gear hardware concept is simple and worth testing. The simulation covers several masses, frequencies, amplitudes, and starting heights, and the code is on GitHub. That is real, reproducible work for what it is.

The soft spots are not hiding. First, Eq. 1 is a resistive-loss proxy; it omits the mechanical power term i*k_e*omega, which does not vanish in cyclic motion, and it omits the energy consumed by the Dynamixel servo and worm gear that adjust the spring. A spring that cancels most of the torque-squared could still leave most of the battery consumption intact. Second, the evaluation is entirely in-sample: the same trajectory generates the fit and the metric. No held-out trajectories, no task variation, no repeat runs, no error bars. The word adaptive overstates it—experiments are offline parameter optimization, not closed-loop adaptation. Third, the claim in Section V that overall energy consumption could be considerably reduced goes beyond what the metric shows.

That said, none of this is fatal to the core idea. The derivation is correct, the hardware concept is worth pursuing, and the deficiencies are addressable with a proper energy model (including mechanical work and actuation cost), held-out evaluation, and repeated trials.

I would send this to a serious referee, not desk reject it. The reader is right that it is incremental relative to refs 6 and 10, but the torsion spring mechanism and the simplicity of the closed-form are enough to justify a review, with major revision expected. For my own work, I would not cite the energy numbers, but I might cite the mechanism concept if I built on it.

Referee Report

4 major / 7 minor

Summary. The paper proposes an adaptive torsion spring mechanism for legged robots, with an adjustable equilibrium position and stiffness, intended to reduce actuator energy consumption during cyclic tasks. The main contribution is a closed-form derivation of optimal spring parameters (stiffness μ* and equilibrium position α0*) that minimize a torque-squared energy proxy (Eqs. 1–4). The method is evaluated in a Gazebo simulation of a 3-DoF leg controlled by a PD controller, comparing a baseline run with a run where a torsion spring with the derived parameters is added. Table I reports energy ratios Ea/E0 between 0.15% and 3.9% over variations in mass, frequency, amplitude, and starting height, and the paper concludes that the overall energy consumption can be considerably reduced.

Significance. If the reported quantitative reductions were valid, the closed-form formulas would provide a straightforward and computationally cheap method for sizing parallel torsion springs in cyclic robotic tasks. The derivation of Eq. (4) is mathematically correct, and the authors make their code and simulation setup available, which supports reproducibility. However, the central claim of significant energy reduction is not established by the evidence presented: the energy metric used is a resistive-loss proxy, the evaluation is in-sample, and the energy cost of the adaptive mechanism itself is omitted. These issues are load-bearing for the paper's main conclusion, so the significance of the contribution as currently stated is limited.

major comments (4)

[Section III-B, Eq. (1) and Table I] The energy metric E = K Σ τ_i² Δt is a torque-squared proxy (proportional to resistive losses in a DC motor) and omits the mechanical power term i k_e ω. In a non-regenerative drive, negative mechanical work is dissipated rather than recovered, so the true electrical energy is not proportional to the sum of squared torques. Consequently, the E_a/E0 ratios reported in Table I do not represent total or even electrical energy consumption, and the conclusion in Section V that 'the overall energy consumption could be considerably reduced' is not supported by this metric.
[Section IV-A and Table I] The optimal spring parameters are computed from the baseline trajectory without the spring, and then energy is measured on that same trajectory with the spring. Since Eq. (4) is the least-squares solution for fitting μ(α - α0) to the baseline torque τ, the ratio E_a/E0 equals 1 - R² of that regression. The reported values (0.15% to 3.9%) therefore quantify the in-sample goodness of fit of a linear spring model to the baseline torque-angle data, not an energy saving. A held-out trajectory, a different gait, or cross-validation is necessary to support any predictive claim about energy reduction.
[Section IV-A and Section IV-B] The paper assumes that the actuator torque as a function of joint angle is unchanged by adding the spring, so that parameters fitted to the baseline remain optimal for the spring-equipped system. However, the spring alters the plant dynamics (it adds torque that changes the acceleration and the closed-loop tracking error), so the actuator torque during the spring run is not necessarily τ_i - μ(α_i - α0). The paper does not verify whether the same parameters would be obtained if the optimization were repeated with the spring present, nor does it quantify trajectory deviations between the baseline and spring-equipped runs.
[Section VI and Figs. 3–4] The energy consumed by the Dynamixel servo and worm gear that adjust and maintain the spring's equilibrium position is not included in Eq. (2) or in Table I. Since this actuation is an integral part of the proposed adaptive mechanism, a fair comparison of total system energy consumption must account for it; otherwise the reported savings could be substantially offset or even reversed.

minor comments (7)

[Keywords] The keyword 'Adaptive Comlpliance' appears to be a typo; it should be 'Adaptive Compliance'.
[Introduction] The phrase 'Part if the recent research' in Section I should be 'Part of the recent research'.
[Section I] The sentence 'The proposed system system relies on a torsion spring' contains a duplicated word 'system'.
[Section V] The conclusion states 'in the experiments with a physical stand the reduction in power consumption will be not as fascinating', but no physical experiments are reported in this paper; the text should clarify that these are simulation results and that a physical test is planned for future work.
[Figure 2 and Table I] The conditions described in the Figure 2 caption (e.g., 'Frequency increased from 100Hz up to 200Hz') do not match the corresponding entries in Table I (which lists periods 0.94, 1.88, 3.77 s). Please align the figure descriptions with the experimental parameters.
[Table I] The table is missing units and clearly defined column headers: μ* is presumably in N·m/rad, α0* in rad, E0 and Ea in (N·m)²·s (or a related unit), and 'h0' is the initial height in meters. Adding units would improve interpretability.
[References] Some references are incomplete or inconsistently formatted, e.g., [3] is a URL-only entry and [17] omits the full author list. Please check journal/conference standards.

Circularity Check

1 steps flagged · score 7.0 of 10

Table I's Ea/E0 ratios are the in-sample residual variance of a two-parameter least-squares fit to the baseline torque-angle data, so the energy reduction is guaranteed by construction rather than independently predicted.

fitted input called prediction [Section III-B, Eqs. (1)-(4); Section IV-B; Table I]
"E = K \sum_{i=1}^T (\tau_i - \mu(\alpha_i - \alpha_0))^2 \Delta t ... Solving this system for \mu and \alpha_0 yields Equations 4. ... For each of the experiments without a spring the optimal \mu^* and \alpha_0^* were calculated. After that, another simulation with a spring was performed."

The spring parameters \mu^* and \alpha_0^* are obtained by minimizing Eq. (2), i.e., by least-squares regression of the baseline actuator torque \tau_i on the knee angle \alpha_i. Under the paper's own model, adding the spring simply subtracts \mu(\alpha_i-\alpha_0) from the actuator torque for the same trajectory, so the 'optimized' energy E_a is exactly the residual sum of squares of that same regression on the same data points used for fitting. Minimization therefore guarantees E_a \le E_0, and the reported ratios E_a/E_0 (as low as 0.15%) are 1 - R^2 of the linear fit, not an out-of-sample prediction of energy savings.

full rationale

The paper's derivation chain is mathematically self-contained but the headline result is circular in a specific, quotable way. The optimal spring parameters are fitted to the baseline trajectory's torque-angle pairs by minimizing exactly the same squared-torque functional that later defines the reported energy consumption. Under the paper's model the spring torque is additive, so the measured E_a must be the residual variance of that fit, making the large reduction in Table I a construction of the optimization rather than an independent empirical finding. The separate simulation with the spring does not break this circularity because it evaluates the same in-sample quantity: the trajectory and the fitted data are the same, and no generalization test is reported. The paper also uses a torque-squared proxy for energy (Eq. 1), omitting mechanical work and the energy consumed by the Dynamixel servo and worm gear used to adjust the spring; while that is a correctness or modeling risk rather than a circularity, it further weakens the inference that 'overall energy consumption' is reduced. There is no load-bearing self-citation or imported uniqueness theorem; the circularity is the fitted-input-called-prediction pattern, which justifies a score of 7.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a torque-squared energy model, an ideal lossless spring, and the assumption that the spring does not change the load torque. These are domain assumptions not verified against hardware or a more detailed motor model.

free parameters (2)

PD gains (Kp, Kd) = Kp=300, Kd=1
Chosen for the position controller; they shape the torque profile from which spring parameters are fitted, so the energy savings depend on them.
Friction coefficient = negligibly small
Set to zero to avoid disturbing motion; this removes dissipative losses that would reduce the relative benefit of the spring.

assumptions (3)

domain assumption Energy consumption is proportional to the sum of squared actuator torques (Eq. 1)
This models only resistive (I^2R) losses and ignores mechanical work and regenerative braking; invoked in Section III-B.
domain assumption A linear torsion spring produces torque μ(α - α0) with no energy cost and without altering the load torque (Eq. 2)
The spring is assumed ideal and the load torque is assumed unchanged when the spring is added; used throughout Section III and IV.
domain assumption The reference trajectory is tracked identically with and without the spring
The PD controller is assumed to enforce the same motion, so the torque reduction is purely the spring compensation; stated in Section IV-A.

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Cite this review

Pith. "Pith review of Optimizing energy consumption for legged robot by adapting equilibrium position and stiffness of a parallel torsion spring." pith.science (2026). https://pith.science/paper/M7AHXHRF

@misc{pith2026241118295,
  author       = {Pith},
  title        = {Pith review of: Optimizing energy consumption for legged robot by adapting equilibrium position and stiffness of a parallel torsion spring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7AHXHRF}},
  note         = {Machine review of arXiv:2411.18295}
}

read the original abstract

This paper is dedicated to the development of a novel adaptive torsion spring mechanism for optimizing energy consumption in legged robots. By adjusting the equilibrium position and stiffness of the spring, the system improves energy efficiency during cyclic movements, such as walking and jumping. The adaptive compliance mechanism, consisting of a torsion spring combined with a worm gear driven by a servo actuator, compensates for motion-induced torque and reduces motor load. Simulation results demonstrate a significant reduction in power consumption, highlighting the effectiveness of this approach in enhancing robotic locomotion.

Figures

Figures reproduced from arXiv: 2411.18295 by the authors.

**Figure 2.** Comparison of knee motor torques during a single [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The experimental setup consists of a leg, fixing stand, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: System overview, including modules and interfaces [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Learning-based design and control for quadrupedal robots with parallel-elastic actuators,

F. Bjelonic, J. Lee, P. Arm, D. Sako, D. Tateo, J. Peters, and M. Hutter, “Learning-based design and control for quadrupedal robots with parallel-elastic actuators,” IEEE Robotics and Automation Letters , vol. 8, no. 3, pp. 1611–1618, 2023

work page 2023

[2] [2]

Realization of nonlin- ear adaptive compliance: Towards energy efficiency in cyclic tasks,

R. Nasiri, A. Ahmadi, and M. N. Ahmadabadi, “Realization of nonlin- ear adaptive compliance: Towards energy efficiency in cyclic tasks,” in 2019 7th International Conference on Robotics and Mechatronics (ICRoM). IEEE, 2019, pp. 175–180

work page 2019

[3] [3]

Variable recruitment of parallel elastic elements: Series-parallel elastic actuators (spea) with dephased mutilated gears,

B. V . Glenn Mathijssen, Dirk Lefeber, “Variable recruitment of parallel elastic elements: Series-parallel elastic actuators (spea) with dephased mutilated gears,” https://researchportal.vub.be/en/publications/ variable-recruitment-of-parallel-elastic-elements-series-parallel, 2014

work page 2014

[4] [4]

Dynamic gravity cancellation and regula- tion control in robots with flexible transmissions: Constant, nonlinear, and variable stiffness,

A. De Luca and F. Flacco, “Dynamic gravity cancellation and regula- tion control in robots with flexible transmissions: Constant, nonlinear, and variable stiffness,” 2010

work page 2010

[5] [5]

An overview on principles for energy efficient robot locomotion,

N. Kashiri, A. Abate, S. J. Abram, A. Albu-Schaffer, P. J. Clary, M. Daley, S. Faraji, R. Furnemont, M. Garabini, H. Geyer et al. , “An overview on principles for energy efficient robot locomotion,” Frontiers in Robotics and AI , vol. 5, p. 129, 2018

2018

[6] [6]

Adaptation in variable parallel compliance: Towards energy efficiency in cyclic tasks,

R. Nasiri, M. Khoramshahi, M. Shushtari, and M. N. Ahmadabadi, “Adaptation in variable parallel compliance: Towards energy efficiency in cyclic tasks,” IEEE/ASME Transactions on Mechatronics , vol. 22, no. 2, pp. 1059–1070, 2016

work page 2016

[7] [7]

Adaptation in a variable parallel elastic actuator for rotary mecha- nisms towards energy efficiency,

O. Mohseni, M. A. Shahri, A. Davoodi, and M. N. Ahmadabadi, “Adaptation in a variable parallel elastic actuator for rotary mecha- nisms towards energy efficiency,” Robotics and Autonomous Systems , vol. 143, p. 103815, 2021

work page 2021

[8] [8]

Compliance and frequency optimization for energy efficiency in cyclic tasks,

M. Shushtari, R. Nasiri, M. J. Yazdanpanah, and M. N. Ahmadabadi, “Compliance and frequency optimization for energy efficiency in cyclic tasks,” Robotica, vol. 35, no. 12, pp. 2363–2380, 2017

work page 2017