Koopman Operator Based Linear Model Predictive Control for Quadruped Trotting

Chun-Ming Yang; Pranav A. Bhounsule

REVIEW 3 major objections 6 minor 24 references

Koopman Operator Based Linear Model Predictive Control for Quadruped Trotting

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A Koopman-lifted linear model trained on an idealized torso-only simulation suffices for real-time linear MPC of a trotting quadruped, as shown by reference tracking and push recovery on physical hardware.

desk verdict First hardware Koopman-LMPC quadruped demo, with a real derivation error and a missing baseline that a serious referee should flag. read the letter →

arxiv 2508.08259 v1 pith:O5M45KYR submitted 2025-07-19 cs.RO

Chun-Ming Yang , Pranav A. Bhounsule This is my paper

classification cs.RO

keywords Koopmanoperatorlinearmodelpredictivecontrolquadrupedtrottingsinglerigidbodyextendeddynamicmodedecompositiondisturbancerejectionreal-time

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 tries to establish that a linear model of a trotting quadruped can be learned by lifting the idealized single-rigid-body dynamics into a 49-dimensional space of observables via the Koopman operator, and that this linear model is accurate enough to serve as the prediction model inside a real-time linear model predictive controller. The authors implement the controller on a physical quadruped and in simulation, reporting that it tracks commanded forward, lateral, and turning speeds and recovers from foot slips and a push-induced pitch deviation of 6.5 degrees. If the claim holds, it means quadruped control can avoid local linearization and hand-tuned gains while still solving a convex quadratic program in about 3 milliseconds, making online optimal control practical without model-free, sample-heavy training. The paper positions this as the first application of Koopman-based linear MPC to quadrupedal locomotion.

What carries the argument

The load-bearing mechanism is the finite-dimensional Koopman operator $K=[A,B]$, computed by extended dynamic mode decomposition from single-rigid-body simulation data. The lifted state is $\Pi = [1, p, \Theta, \dot{p}, \Omega, \mathrm{vec}(R\Omega), \ldots, \mathrm{vec}(R\Omega^4)]^\top \in \mathbb{R}^{49}$, where $R$ is the torso rotation matrix, $\Omega$ is the body angular velocity, and the constant 1 absorbs gravity; the rotation-matrix observables preserve the rotational nonlinearity that a first-order linearization would drop. The LMPC evolves $\Pi$ linearly, pulls the physical torso state back out with a selection matrix, and solves a quadratic program for ground reaction forces, with joint torques obtained through the leg Jacobian.

What would settle it

Collect hardware trotting data under rich excitation, fit the same 49-observable extended dynamic mode decomposition from those data, and compare one-step prediction errors on held-out hardware roll-outs with the model fitted to simulated single-rigid-body data; if the two are not of the same order of magnitude, the claim that simulated SRB data suffice for hardware linear MPC is refuted.

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Extended reading notes

Core claim

The paper's central claim is that the nonlinear single rigid body (SRB) dynamics of a trotting quadruped can be embedded, through Koopman operator theory, into a finite-dimensional linear model that is accurate enough for linear model predictive control on hardware. Using a lifted state of 49 observables (the torso state plus vectorized products of the rotation matrix with powers of the body angular velocity) and fitting the operator by extended dynamic mode decomposition on simulated SRB data, the authors obtain matrices $A$ and $B$ whose prediction error stays within $\pm 10^{-3}$ over 0.05 s roll-outs. This linear model is embedded in a constrained quadratic program solved at a 200 Hz update rate, and the paper reports that the robot follows velocity commands with small root mean square error and recovers from a 6.5-degree pitch disturbance and foot slips. The authors state that this is the first use of Koopman operator theory for linear MPC of quadrupedal locomotion.

Load-bearing premise

The load-bearing premise is that the idealized model of a heavy torso on massless legs represents the real robot's trotting well enough that a linear model trained only on simulated data stays valid on hardware; if leg dynamics, foot slip, compliance, or actuator delay matter more than assumed, the reported disturbance rejection would degrade.

Editorial extensions

If this is right

The same simulation-only identification recipe should transfer to other quadrupeds whose dynamics are well approximated by the single-rigid-body model, since no hardware data or per-robot tuning is needed.
A 6 ms horizon solved in about 3 ms leaves computation budget at the 200 Hz update rate for longer horizons, more constraints, or additional estimation work.
The authors report that prediction error grows beyond about 0.05 s, so the short receding horizon is load-bearing; extending it would require a larger dictionary or a bilinear Koopman realization.
The reported closed-loop RMSE values (about 0.026 m/s for velocity steps and 0.018 rad for orientation under push) provide concrete benchmarks for future Koopman-based legged controllers.

Reading between the lines

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

A direct ablation against conventional linearized-SRB MPC on the same hardware tests would separate the contribution of the Koopman lift from that of the MPC formulation; the paper does not include this comparison.
Fitting the same dictionary to actively excited hardware data and checking agreement with the SRB-fitted matrices would be a sharper sim-to-real validation than the closed-loop experiments alone.
Because the lifted dictionary contains only torso observables, adding swing-leg or contact-event observables could extend the valid horizon and disturbance envelope beyond the moderate pushes tested.
The same SRB-to-Koopman recipe should carry over to other periodic gaits, but the contact schedule in the finite state machine would need to change; the authors list multiple gaits as future work.

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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

First, the thing you should know: this is a genuine hardware demonstration of Koopman-based LMPC on a quadruped, the first I know of. The robot tracks reference speeds and recovers from pushes. That is the paper's solid core. It is not a simulation-only paper or a purely methodological increment.

What's new and good: they fit a 49-observable Koopman model to SRB rollouts, then use it in LMPC solved with qpSWIFT at 3 ms. The hardware results are meaningful because the model was fitted entirely on simulation data, so the push-recovery and tracking results on the Go1 are independent evidence that the SRB-plus-Koopman model is adequate for moderate conditions. No code or data is released, but the video link and RMSE numbers give a reasonable basis for evaluation.

Soft spots, in order. The QP derivation in Eq. 14 is wrong as written. Equations 12-13 say the lifted state evolves linearly and the original state is recovered by Cx. Recovering the original state and then re-lifting is a nonlinear operation. You cannot get X_qp = A_qp x0 + B_qp U_qp for arbitrary x0 unless x0 is already the lifted state or the dictionary is linear. This is not a nit. It is the core derivation of the controller. If the implementation actually lifts x0 before forming the QP, the equations need to say that; the paper doesn't. The stress-test note is correct.

Second, there is no linearized-SRB LMPC baseline. The stated motivation is avoiding linearization error, but they never show that Koopman LMPC beats a linearized LMPC. The hardware tracking numbers are decent, but without a baseline it's unclear what the Koopman model buys you.

Third, Fig. 4 is largely a fitting residual: EDMD model trained on SRB rollouts and evaluated on SRB rollouts. It shows the fit is good, not that the model generalizes. The hardware results are the real external validation, and they're the best part of the paper.

Minor: "preserves the nonlinearity" is an overstatement for a finite-dimensional EDMD model. Also, the SRB idealization is a real assumption; it works in their tested conditions, but they don't validate the model against hardware data.

Who this is for: people working on Koopman MPC or legged locomotion control. It deserves a serious referee, but the referee should demand a corrected derivation and a baseline comparison. I'd accept for peer review with major revision.

Referee Report

3 major / 6 minor

Summary. The paper proposes a Koopman-operator-based linear model predictive control (LMPC) for quadruped trotting. The nonlinear single rigid body (SRB) model of a quadruped is lifted via a physics-informed dictionary of observables, and extended dynamic mode decomposition (EDMD) is used to fit a finite-dimensional linear predictor. A quadratic program is then formulated from this lifted linear model and solved online to generate ground reaction forces for stance legs. The controller is evaluated in MuJoCo simulation and on a Unitree Go1 hardware platform for reference tracking, slip recovery, and push disturbance rejection. The paper claims this is the first use of Koopman operator theory for LMPC of quadrupedal locomotion.

Significance. If the technical derivation is corrected and the claims are properly supported, this would be a useful demonstration that a low-dimensional, data-driven linear model can replace a locally linearized model in real-time quadruped control. The hardware experiments are a genuine strength: they show real-time closed-loop control with a 3 ms QP solve and moderate disturbance rejection. The use of a 49-dimensional Koopman dictionary derived from the SRB model is a sensible design choice that keeps the lifted dimension small. However, the paper's central derivation has a coordinate-mixing issue in the QP formulation, and the experiments lack a linearized-MPC baseline, so the motivating claim that Koopman LMPC avoids linearization errors is not directly tested.

major comments (3)

[§II-D, Eq. (14)] Equation (14) does not follow from the lifted dynamics defined in Eqs. (12)-(13). Since the dictionary Π in Eq. (10) contains nonlinear observables such as RΩ^p and the constant 1, the first predicted state is x_1 = C_x (A Π(x_0) + B u_0), and the k-step prediction is an affine function of Π(x_0), not of x_0. Replacing Π(x_0) with x_0 in Eq. (14) changes the initial condition of the QP, and therefore the matrices H and P in Eq. (16) are not the ones implied by the Koopman model. Please correct Eq. (14) to operate on Π(x_0), with an appropriate block-diagonal extraction matrix, and revise H and P accordingly; alternatively, state explicitly that the implementation first lifts x_0 and provide the matching derivation.
[§III, experimental comparison] The abstract and introduction motivate the approach by arguing that local linearization of the equations of motion introduces model inaccuracies, but none of the simulation or hardware experiments compare the proposed Koopman LMPC against a conventional linearized SRB LMPC or another baseline controller. The reported RMSE values against reference trajectories do not by themselves show that the Koopman formulation preserves nonlinearity better than a linearized model. Please add at least a simulation baseline with a standard linearized SRB MPC under the same conditions, and ideally a hardware comparison, to support the central claim.
[§II-C, §III-A, §IV] The Koopman model is identified from SRB rollouts and then evaluated on SRB rollouts in Fig. 4, which is essentially a check of the fitting accuracy on the training distribution. The paper never validates the learned Koopman predictor against hardware state trajectories. Because the hardware controller relies entirely on the SRB idealization, the conclusion that the model is 'adequate' for hardware trotting rests on the untested assumption that the SRB model captures the real Go1 dynamics under slips and pushes. Please validate the Koopman model on hardware data, or at least quantify its prediction error on measured state trajectories and discuss the implications for the hardware results.

minor comments (6)

[§II-D, Eq. (12)] The recurrence in Eq. (12) has an index inconsistency: it should read Π_{t+i+1} = A Π_{t+i} + B u_{t+i} (or equivalent), not 'Π_{t+i} = AΠ_i + Bui'.
[§II-D, Eq. (13)] The expression x_i = C_x Π(x_i) is self-referential; it should define x_i as C_x Π(x_i) where the subscript on the left is the time index and the argument of Π is the corresponding state, but the notation should be cleaned up to avoid circularity.
[§II-D, Eq. (16)] The constraint 'c ≤ CU_qp ≤ c' appears to use the same symbol for upper and lower bounds; presumably the lower and upper bound vectors are distinct. Please correct the notation.
[§II-E.3] The sentence 'The planning horizon for the model predictive control is 6 ms or 166.67 Hz' is dimensionally unclear: 6 ms is a time duration, while 166.67 Hz is a frequency. Please specify whether the horizon is 6 ms, 6 control steps, or something else, and correct the update-horizon description as well.
[§I, References] Reference [13] is a 2001 MPC paper for autonomous vehicles; it does not appear to use Koopman operator theory and seems out of place in the list of Koopman applications. Please either replace it with a relevant Koopman-based vehicle control reference or remove it.
[§I] There is a minor grammatical error: 'Deep Reinforcement learning (DRL) is an model-free method' should be 'is a model-free method.'

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Koopman model is fit to SRB simulation data, but the central LMPC claim is evaluated on independent MuJoCo and hardware experiments; only a non-circular derivation gap in Eq. 14 is noted.

full rationale

The paper's central claim is that a Koopman-linearized SRB model can support real-time LMPC for quadruped trotting, with hardware tracking and disturbance rejection as the primary evidence. The Koopman matrices are identified via EDMD from SRB simulation rollouts (Sec. II-C), while the evaluation that supports the central claim comes from the full-body MuJoCo simulation (Sec. III-B) and physical hardware experiments (Sec. III-C). Neither of these evaluation targets is used to fit any parameter, so the main result is not circular. The Fig. 4 model-fit check generates both the 'actual' and the Koopman-predicted trajectories from the same SRB model, but the test rollouts are new initial conditions and force profiles that were not used in the EDMD regression; this is a standard held-out, in-distribution validation of the learned model rather than a prediction forced by the fitting data. The self-citation to the authors' prior work [1] is used only for the standard analytical inverse kinematics of the swing-leg controller and is not load-bearing for the Koopman or LMPC contribution. No uniqueness theorem, ansatz-smuggling via self-citation, or renaming of a known result is present. A separate, non-circular concern is that Eq. 14 in Sec. II-D writes X_qp = A_qp x0 + B_qp U_qp even though Eqs. 12-13 define the lifted prediction starting from the nonlinear dictionary Pi(x0), not from the raw state x0; as written, the QP does not formally follow from the Koopman model. This is a derivation-chain gap and a correctness risk, but it is not an equivalence of a result to its input, so it does not raise the circularity score.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the SRB idealization, the choice of lifting dictionary, and the EDMD fitting procedure. The paper introduces no new physical entities, but it does introduce several hand-chosen parameters (dictionary order, MPC horizon, weights, gains) and relies on the assumption that a model trained on simulated SRB data transfers to hardware without adaptation.

free parameters (6)

Dictionary order p (number of RΩ moments) = 4
Hand-chosen; determines the lifted state dimension (49 observables); no ablation or convergence study is provided.
MPC planning horizon and update rate = 6 ms horizon, 5 ms update (200 Hz)
Chosen to fit the real-time QP solve time of about 3 ms; no analysis of horizon sensitivity.
LMPC weight matrices Q and R = Not specified numerically
Described as user-chosen diagonal positive definite matrices; exact values are not given, affecting reproducibility.
Control input bounds umin and umax = Not specified
Constraints on ground reaction forces are assumed but never quantified.
Number of training rollouts for EDMD = 100
Chosen by the authors; no sensitivity analysis for the number of rollouts.
Swing controller PD gains Kp, Kd = Not specified
Used in Eq. 18 for swing leg torque, but gains are not given.

assumptions (5)

domain assumption The SRB model with massless legs is an adequate representation of the quadruped's dynamics for trotting.
Invoked in Sec. II-A and Sec. IV; the paper relies on this idealization to generate training data and to design the controller.
ad hoc to paper The chosen finite dictionary of observables Π = [1, p, Θ, ṗ, Ω, RΩ, ..., RΩ^4] adequately linearizes the nonlinear SRB dynamics over the operating region.
Sec. II-C selects these physics-informed observables, but no convergence or error bound is provided; the paper acknowledges accuracy diminishes over time.
standard math EDMD with random initial conditions and random force profiles produces a valid linear model of the SRB dynamics.
Sec. II-B uses least-squares regression; this is a standard approach, but the quality depends on the excitation and dictionary, which are not analyzed.
domain assumption Stance leg ground reaction forces can be commanded directly through joint torques via the Jacobian transpose (Eq. 19).
Assumes no significant actuator dynamics or foot slip that would break the force-torque mapping.
domain assumption A 6 ms MPC planning horizon is sufficient for stable trotting and disturbance rejection.
Supported only empirically; no analysis of the horizon relative to the 0.2 s gait cycle.

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Cite this review

Pith. "Pith review of Koopman Operator Based Linear Model Predictive Control for Quadruped Trotting." pith.science (2026). https://pith.science/paper/O5M45KYR

@misc{pith2026250808259,
  author       = {Pith},
  title        = {Pith review of: Koopman Operator Based Linear Model Predictive Control for Quadruped Trotting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5M45KYR}},
  note         = {Machine review of arXiv:2508.08259}
}

read the original abstract

Online optimal control of quadruped robots would enable them to adapt to varying inputs and changing conditions in real time. A common way of achieving this is linear model predictive control (LMPC), where a quadratic programming (QP) problem is formulated over a finite horizon with a quadratic cost and linear constraints obtained by linearizing the equations of motion and solved on the fly. However, the model linearization may lead to model inaccuracies. In this paper, we use the Koopman operator to create a linear model of the quadrupedal system in high dimensional space which preserves the nonlinearity of the equations of motion. Then using LMPC, we demonstrate high fidelity tracking and disturbance rejection on a quadrupedal robot. This is the first work that uses the Koopman operator theory for LMPC of quadrupedal locomotion.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A simple controller for omnidirectional trotting of quadrupedal robots: Command following and waypoint tracking,

P. A. Bhounsule and C.-M. Yang, “A simple controller for omnidirectional trotting of quadrupedal robots: Command following and waypoint tracking,” Robotics, vol. 12, no. 2, 2023. [Online]. Available: https://www.mdpi.com/2218-6581/12/2/35

2023

[2] [2]

Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge,

P. A. Bhounsule, J. Cortell, A. Grewal, B. Hendriksen, J. D. Karssen, C. Paul, and A. Ruina, “Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge,” International Journal of Robotics Research , 2014

work page 2014

[3] [3]

Highly dynamic quadruped locomotion via whole-body impulse control and model predictive control,

D. Kim, J. Di Carlo, B. Katz, G. Bledt, and S. Kim, “Highly dynamic quadruped locomotion via whole-body impulse control and model predictive control,” arXiv preprint arXiv:1909.06586 , 2019

arXiv 1909

[4] [4]

Representation-free model predictive control for dynamic motions in quadrupeds,

Y . Ding, A. Pandala, C. Li, Y .-H. Shin, and H.-W. Park, “Representation-free model predictive control for dynamic motions in quadrupeds,” IEEE Transactions on Robotics , vol. 37, no. 4, pp. 1154–1171, 2021

work page 2021

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M. Neunert, M. St ¨auble, M. Giftthaler, C. D. Bellicoso, J. Carius, C. Gehring, M. Hutter, and J. Buchli, “Whole-body nonlinear model predictive control through contacts for quadrupeds,” IEEE Robotics and Automation Letters , vol. 3, no. 3, pp. 1458–1465, 2018

2018

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G. Feng, H. Zhang, Z. Li, X. B. Peng, B. Basireddy, L. Yue, Z. Song, L. Yang, Y . Liu, K. Sreenathet al., “Genloco: Generalized locomotion controllers for quadrupedal robots,” arXiv preprint arXiv:2209.05309 , 2022

arXiv 2022

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S. Chen, B. Zhang, M. W. Mueller, A. Rai, and K. Sreenath, “Learning torque control for quadrupedal locomotion,” in 2023 IEEE-RAS 22nd International Conference on Humanoid Robots (Humanoids) . IEEE, 2023, pp. 1–8

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Dtc: Deep tracking control,

F. Jenelten, J. He, F. Farshidian, and M. Hutter, “Dtc: Deep tracking control,” Science Robotics , vol. 9, no. 86, p. eadh5401, 2024

work page 2024