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REVIEW 4 major objections 5 minor 34 references

Benchmarking Different QP Formulations and Solvers for Dynamic Quadrupedal Walking

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that in dynamic quadrupedal walking, the sparse MPC formulation solved with HPIPM speed abs is fastest on all tested hardware, and the ARM-based Jetson Orin is the most energy-efficient platform.

desk verdict Useful engineering benchmark whose solver and hardware recommendations are not yet fully supported: the solver ranking ignores solution accuracy, and the power metric is sampled too coarsely. read the letter →

arxiv 2502.01329 v1 pith:5L6ON44L submitted 2025-02-03 cs.RO

Franek Stark , Jakob Middelberg , Dennis Mronga , Shubham Vyas , Frank Kirchner This is my paper · ORCID
classification cs.RO
keywords quadraticprogrammingmodelpredictivecontrolwhole-bodyquadrupedalwalkingsparseformulationssolverbenchmarksolvefrequencyperwattembeddedcomputing
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

This paper sets out to give practitioners a defensible default choice for the quadratic programs (QPs) at the heart of quadrupedal walking control. It claims that for model predictive control (MPC), the sparse problem formulation solved with the interior-point solver HPIPM in its speed-focused mode is the fastest option across every tested hardware platform, condensing level, and horizon. For whole-body control (WBC), it claims solver choice matters little; the formulation and setup dominate. It also introduces a metric, solve frequency per watt, and reports that the ARM-based embedded platform outperforms both x86 systems in energy efficiency for this workload, making it the recommended choice for battery-limited robots.

What carries the argument

Two mechanisms carry the argument. First, the SFPW metric, defined as $(\text{solve time})^{-1}$ divided by CPU power consumption, with CPU power sampled at 10 Hz through Intel RAPL on x86 and tegrastats on ARM; it converts raw solve times into a hardware-independent efficiency ranking. Second, the partial-condensing formulation, which splits the prediction horizon N into Np blocks and eliminates state variables inside each block, so the benchmark can sweep continuously from fully sparse to fully dense QPs. The solver set spans interior-point, active-set, ADMM/ALM, and dual active-set methods, with HPIPM providing both the sparse solver and the condensing routines.

What would settle it

Run the same benchmark with power measured at kilohertz rate or with direct current sensing and compare the SFPW ranking; if the Jetson Orin no longer gives roughly 2-3x the efficiency of the desktop or LattePanda, the hardware-efficiency claim fails. Separately, finding any condensing level and horizon on one tested computer where a solver other than HPIPM speed abs achieves the lowest mean solve time would falsify the solver-dominance claim.

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Extended reading notes

Core claim

The paper's central claim is that a single solver and formulation combination dominates for MPC on walking robots: the sparse QP formulation using HPIPM's speed-abs interior-point mode. In their experiments this combination delivered the lowest mean solve time for all three computers, for both horizons (N=10 and N=20), and across all partial-condensing levels; dense formulations became impractical at N=20 for several solvers. For WBC, all tested solvers stayed below 0.5 ms, with Eiquadprog fastest in every comparison, and the larger full TSID formulation increases time only slightly. The paper further claims that the Jetson Orin is the most efficient platform: more than twice as efficient as the LattePanda and about three times as efficient as the desktop PC for MPC, and roughly four times as efficient for WBC, as measured by the new solve-frequency-per-watt metric.

Load-bearing premise

The efficiency ranking rests on CPU power measurements sampled at 10 Hz being accurate for QP solves that take microseconds to milliseconds; a noisy or background-dominated power sample could flip the ordering between platforms.

Editorial extensions

If this is right

  • MPC users on legged robots can default to a sparse formulation with HPIPM's speed-abs mode rather than hand-tuned dense QPs, especially at longer horizons.
  • For WBC, engineering effort is better spent on the problem formulation than on solver selection, since all tested solvers are comfortably fast.
  • Energy-constrained deployments should favor ARM-based embedded boards such as the Jetson Orin, which delivered roughly 2-3x the solve frequency per watt of x86 systems.
  • At short horizons (N=10), active-set solvers like qpOASES and DAQP are competitive, so dense formulations remain viable for smaller problems.
  • Interior-point solvers show more stable solve-time distributions across changing contact schedules, which matters for dynamic gait.

Reading between the lines

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

  • The 10 Hz power sampling may undercount brief solve bursts; if so, the ARM-efficiency conclusion could change under a higher-rate power measurement, so the hardware ranking should be re-tested before being treated as universal.
  • The benchmark uses standard solver hyperparameters and a simulated trotting and standing task; per-application tuning or real-hardware power draw could shift the ranking.
  • Because the MPC problem has input dimension close to state dimension, sparse wins; the paper's own reasoning implies condensing could win for robots with far fewer actuators, a testable prediction.
  • The SFPW metric could be adopted more broadly as a standard for comparing control algorithms on embedded robots, not just QP solvers.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript benchmarks QP formulations and solvers for MPC and WBC in dynamic quadrupedal walking on three computing platforms (Jetson Orin NX, LattePanda Alpha, Desktop PC). It compares sparse, partially condensed, and fully condensed MPC formulations across prediction horizons N=10 and N=20, and two WBC variants (reduced and full TSID), using solvers HPIPM (balanced and speed_abs), OSQP, qpOASES, DAQP, PROXQP, and Eiquadprog. It introduces Solve Frequency per Watt (SFPW) as a cross-hardware efficiency metric. Experiments use a simulated Unitree Go2 in Drake with a standard MPC+WBC controller in trotting and standing scenarios. Main findings are that HPIPM speed_abs is fastest across all architectures, condensing levels, and horizons; sparse formulations outperform dense ones for MPC; all tested WBC solvers perform adequately; and the Jetson Orin is the most power-efficient platform. The benchmark code is made open source.

Significance. The paper addresses a practical gap: there is no systematic comparison of QP formulations, solvers, and hardware for legged MPC/WBC. If the findings are robust, they give actionable defaults to practitioners and highlight that WBC formulation choice matters more than solver choice. Strengths include the reproducible open-source implementation (controller plus benchmark), coverage of partial condensing levels via acados, inclusion of multiple solver methods (IPM, ADMM, ASM, ALM), and a novel efficiency metric (SFPW) that, despite measurement issues, is a useful step. The qualitative trends (sparse better for longer horizons, IPM robust to changing problem structure) are plausible and consistent with prior work. However, the quantitative claims require stronger experimental controls, especially regarding solution accuracy and power measurement.

major comments (4)
  1. [§III-B, §IV-A] The headline result that "For all computer architectures, condensing levels, and horizons, the best-performing solver is HPIPM speed abs" is based exclusively on wall-clock solve time, with no measure of solution quality (e.g., KKT residual, primal/dual constraint violation, or cost gap). The manuscript itself states that HPIPM's balanced mode "provides more accurate results" while speed_abs "focuses on speed" (Section III-B) and later concedes that speed_abs "does not provide accurate results for some applications" (Section V). Because speed_abs is admitted to trade away accuracy, its speed advantage may be an artifact of looser termination tolerances rather than a genuine algorithmic gain. The robot-falling failure criterion is too coarse to detect inaccurate but stabilizing MPC solutions. Please report accuracy metrics per solver and mode, and either restrict the ranking to solutions meeting a common accuracy threshold or discuss how the speed-accuracy tradeoff affects each recommendation.
  2. [§III-A, §IV-C, Table IV] The SFPW metric divides mean solve time by CPU power consumption sampled at 10 Hz, while individual QP solves take microseconds to milliseconds. As a result, the power sample may be dominated by background load or miss the solve burst entirely, so the cross-platform efficiency conclusion ("Jetson Orin performs best... more than twice as efficient as the LattePanda and around three times as efficient as the desktop PC") is not robust to measurement error. No error bars or confidence intervals are provided for the SFPW values in Table IV. Please either synchronize power measurement with the benchmark workload, report integrated energy over a fixed number of solves, or provide a sensitivity analysis showing that the ranking is stable under plausible power-measurement offsets.
  3. [§IV-A, Fig. 3] The reported "mean MPC solve time" includes condensing time, stated as "less than 1 ms." Since some of the fastest solves are sub-millisecond, condensing time can be a large fraction of the reported total; moreover, dense-interface solvers (qpOASES, DAQP) necessarily include different preprocessing than sparse-interface solvers with partial condensing. The paper therefore conflates formulation/condensing cost with solver performance. Please report solver-only time and condensing/setup time separately for each condensing level, and report distributions (e.g., percentiles) in addition to means.
  4. [§III-D, §IV] The experimental protocol lacks statistical detail: no number of trials per configuration, no error bars, and no explicit handling of failed runs beyond marking "Exp. failed" and the two-retry rule if the robot falls. This matters because several comparisons are described as marginal (e.g., HPIPM balance vs. OSQP on x86 for N=20) or as "similar" (Jetson vs. LattePanda), yet no dispersion or significance measure is given. Please add per-configuration repetitions with standard deviations or percentiles and state the number of trials.
minor comments (5)
  1. [§I] The sentence "the impact of QP formulation ... on performance used has yet to be comprehensively examined" is grammatically broken and hard to parse; please revise.
  2. [§IV-A] The sentence "both qpOASES and DAQP perform almost as well as OSQP with Np = 20 (sparse problem)" appears to be a typo; for the N=10 comparison, the sparse problem has Np=10, not Np=20.
  3. [Table IV] Several numeric entries contain stray spaces (e.g., "8 .82", "67.30", "753 .73"); please fix the formatting.
  4. [§II-B, Eq. (2e)] The text says "we use the subscript a to account only for the actuated joints," but the equation notation tau_m, tau_M is not subscripted; please clarify the relationship between S, tau, and the torque limits.
  5. [Fig. 3] The red circles marking failed experiments may be hard to see in grayscale; consider using distinct markers or adding a legend entry that remains legible in print.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solver ranking and hardware-efficiency conclusions rest on direct measurements, not on fitted parameters or self-referential derivations.

full rationale

The paper's central claims are benchmark measurements: mean solve times for QP solvers across hardware (Fig. 3, Table IV) and an efficiency metric SFPW defined as reciprocal solve time divided by CPU power (Eq. 3). The solver ranking ('the best-performing solver is HPIPM speed abs', Section IV-A) is a direct reading of measured solve times; HPIPM speed_abs is a preset mode of an external solver, not a parameter fitted in this paper. The hardware-efficiency conclusion ('the Jetson Orin performs best', Section IV-C) follows from the SFPW values in Table IV, which are computed from independently measured solve times and power readings; the metric is a stated evaluation criterion (inspired by Green500), not a quantity defined in terms of the conclusion. Self-citations to the authors' ARC-OPT framework [26], [27] are implementation references for the WBC benchmark harness; the WBC results are measured with that open-source framework and do not use the citations to justify the solver ranking or to forbid alternatives. The paper's own caveat that HPIPM speed_abs 'does not provide accurate results for some applications' (Section V) is a limitation on validity of the recommendation, not a circularity: no accuracy metric is fitted and then renamed as a prediction. No equation is equivalent to another by construction, and no fitted parameter is relabeled as an independent finding. Therefore no circular step is identified.

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

The paper introduces SFPW, a new metric, but this is a definition rather than an invented physical entity. No free parameters are fitted; all conclusions rest on measured solve times, the 10 Hz power sample, and the representativeness of the benchmark scenarios.

assumptions (3)
  • domain assumption The linearized Single Rigid Body Dynamics (SRBD) model is an adequate prediction model for dynamic quadruped MPC.
    MPC formulation in Eq. (1b) uses a linearized SRBD model around the current state; if this model is too inaccurate, the benchmark's MPC problem is not representative of production dynamic walking. Section II-A.
  • domain assumption CPU power consumption sampled at 10 Hz is a faithful proxy for energy use during microsecond-to-millisecond QP solves.
    SFPW divides by CPU power sampled at 10 Hz (Section III-A); the power measurement may include unrelated load or miss short solve bursts, and no whole-system power is measured.
  • domain assumption Default hyperparameters for all QP solvers are a fair basis for comparison.
    Section III-B states all solvers use standard hyperparameters; tuning could change rankings, so the conclusions apply to out-of-the-box configurations.

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Pith. "Pith review of Benchmarking Different QP Formulations and Solvers for Dynamic Quadrupedal Walking." pith.science (2026). https://pith.science/paper/5L6ON44L

@misc{pith2026250201329,
  author       = {Pith},
  title        = {Pith review of: Benchmarking Different QP Formulations and Solvers for Dynamic Quadrupedal Walking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5L6ON44L}},
  note         = {Machine review of arXiv:2502.01329}
}
read the original abstract

Quadratic Programs (QPs) are widely used in the control of walking robots, especially in Model Predictive Control (MPC) and Whole-Body Control (WBC). In both cases, the controller design requires the formulation of a QP and the selection of a suitable QP solver, both requiring considerable time and expertise. While computational performance benchmarks exist for QP solvers, studies comparing optimal combinations of computational hardware (HW), QP formulation, and solver performance are lacking. In this work, we compare dense and sparse QP formulations, and multiple solving methods on different HW architectures, focusing on their computational efficiency in dynamic walking of four legged robots using MPC. We introduce the Solve Frequency per Watt (SFPW) as a performance measure to enable a cross hardware comparison of the efficiency of QP solvers. We also benchmark different QP solvers for WBC that we use for trajectory stabilization in quadrupedal walking. As a result, this paper provides recommendations for the selection of QP formulations and solvers for different HW architectures in walking robots and indicates which problems should be devoted the greater technical effort in this domain in future.

Figures

Figures reproduced from arXiv: 2502.01329 by the authors.

Figure 1
Figure 1. Go2 quadruped [6] used for evaluation in a Volcanic field test in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Block diagram showing dynamic walking controller’s architecture [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Mean MPC solution time for all solvers, target platforms and condensation levels for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: Computation time for different solvers using the [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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