Koopman Operator Based Linear Model Predictive Control for 2D Quadruped Trotting, Bounding, and Gait Transition

Chun-Ming Yang; Pranav A. Bhounsule

REVIEW 5 major objections 6 minor 28 references

Koopman Operator Based Linear Model Predictive Control for 2D Quadruped Trotting, Bounding, and Gait Transition

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Using Koopman lifting, this paper builds linear models of a quadruped in a high-dimensional space and shows that a single linear MPC formulation produces trotting, bounding, and both transitions in a MuJoCo simulation of the Unitree Go1…

desk verdict Koopman-EDMD LMPC for a simulated quadruped: a real application, but the 'retains nonlinearity' claim does not survive the constant-B input coupling. read the letter →

arxiv 2507.14605 v1 pith:MPPKOL56 submitted 2025-07-19 cs.RO

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

classification cs.RO

keywords Koopmanoperatorextendeddynamicmodedecompositionlinearmodelpredictivecontrolquadrupedlocomotiongaittransitiontrottingboundinghybridsystems

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 claims that Koopman operator theory can replace the usual per-step linearization inside model predictive control for quadrupedal robots. It builds linear models of the robot's single-rigid-body equations in a high-dimensional lifted space, with separate models for the stance and flight phases, and feeds them to a standard linear MPC that runs online. In the MuJoCo simulation of the Unitree Go1, this produces stable trotting and bounding gaits, smooth trot-to-bound and bound-to-trot transitions, and rough-terrain operation, with the quadratic program solved in about 3 ms. The payoff, if the claim holds, is that a single linear controller can manage multiple gaits and their transitions without re-linearizing the equations of motion at every step.

What carries the argument

The load-bearing object is the finite-dimensional Koopman predictor, obtained by lifting the single-rigid-body state into a physics-informed dictionary of observables $\Pi(x) = [1, p, \theta, \dot{p}, \dot{\theta}, R\dot{\theta}, R\dot{\theta}^2, \ldots] \in \mathbb{R}^{7+4p}$, where the rotation-matrix terms are chosen to match the structure of the single-rigid-body equations. Extended Dynamic Mode Decomposition regresses constant matrices $A$ and $B$ from random roll-outs so that $\Pi(x_{t+1}) = A\Pi(x_t) + B u_t$; one such pair is learned for each contact mode (trot stance, front-stance bound, rear-stance bound, and flight). The linear MPC then optimizes a quadratic cost over this lifted model, recovers the physical state through a selection matrix $C_x$, and is solved online by qpSWIFT at 166.67 Hz, while a finite-state machine and PD leg controllers execute the resulting ground-reaction forces.

What would settle it

Reproduce the model-fitting experiment of the paper but measure the Koopman prediction error against the full MuJoCo torso states, rather than the single-rigid-body integration, over the same 0.05 s horizon; if the error is substantially larger than the reported $\pm10^{-4}$ band, or if trotting and bounding become unstable when leg-swing dynamics are removed from the training set, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the nonlinear 2D single-rigid-body dynamics (torso as one rigid body, legs as force sources) of a quadruped can be lifted into a finite-dimensional linear system by the Extended Dynamic Mode Decomposition, and that the resulting hybrid predictor, separate Koopman models for the stance modes and the aerial phase, is accurate enough (prediction errors within $\pm10^{-4}$ over 0.05 s roll-outs) to drive an online Linear Model Predictive Controller. Using that controller, the authors report stable trotting with forward-velocity RMSE 0.029, bounding with RMSE 0.189, and successful trot-to-bound and bound-to-trot switches on both level and rough terrain, with the bound-to-trot switch deliberately placed in the flight phase to avoid a large pitch impulse. The paper presents this as the first demonstration of Koopman-based models for a hybrid quadruped system capable of online multi-gait and gait-transition generation.

Load-bearing premise

The load-bearing premise is that the 2D single-rigid-body model, which is the only data source for training the Koopman models, faithfully represents the full Unitree Go1 dynamics in MuJoCo, including leg-swing effects during flight and contact; the paper provides no direct comparison between single-rigid-body predictions and the full simulation.

Editorial extensions

If this is right

The same lifted linear structure can be applied to other periodic gaits, such as pacing or pronking, by training a Koopman model for each contact mode and scheduling them through the finite-state machine.
Because the online solve takes about 3 ms, the approach should run on embedded hardware at 100+ Hz without specialized solvers, given a sufficiently trained dictionary.
Gait transitions reduce to switching between pre-trained linear models at the correct phase, rather than recomputing a full hybrid trajectory, which keeps the transition cost the same as steady-state control.

Reading between the lines

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

One extension the authors do not develop: because each contact mode has its own linear model, a terrain-aware scheduler could replace the contact-sensor foot-holding heuristic with online mode selection, potentially improving rough-terrain performance.
The authors' stated accuracy decay beyond 0.05 s suggests that the fixed physics-informed dictionary is the bottleneck; augmenting it with neural-network or bilinear terms, as the discussion suggests, could extend the prediction horizon without abandoning the linear-MPC framework.
A testable next step would be to add swing-leg dynamics and compliance into the EDMD training data; if the same controller then works on hardware, it would confirm that the single-rigid-body training source, rather than the Koopman lifting, is the main simplification.
The sixfold gap between trot and bound RMSE (0.029 vs 0.189) is an indirect sign that the current observable dictionary captures contact-rich, low-flight dynamics better than the flight-heavy bounding gait; testing richer dictionaries directly on bounding would clarify this.

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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

Here's my take on arXiv:2507.14605.

The paper's real contribution is narrow but genuine: it appears to be the first application of phase-dependent Koopman-EDMD linear models to a quadruped, with online LMPC demonstrated for trotting, bounding, and gait transitions in MuJoCo. The math is standard EDMD; there is no new theory. What is good: the model fit is checked against held-out SRB rollouts, the simulation results are reasonably clean, and the authors are honest in Section IV that accuracy degrades beyond 0.05 s and that a bilinear Koopman model would better capture the control-affine SRB structure.

Now the soft spots, in rough order of severity. First, the central claim that this 'retains the nonlinearity of the EOM' is not established. The SRB has R(theta)F in Eq. (1), a state-dependent input gain, while the lifted model (Eq. 11) uses a constant B. A constant B cannot represent that product; EDMD can only fit an average input gain over the training distribution. The authors' own Section IV admission about bilinear models confirms the limitation. A targeted test at nonzero pitch or a direct check of input-output coherence would be needed to justify the claim.

Second, there is no baseline comparison. We never see how this compares against the standard linearized SRB MPC (Di Carlo et al.) or a simple LMPC that linearizes at the current state. That is a fixable omission, but important.

Third, the SRB-to-simulation gap is unexamined. The Koopman model is trained on SRB trajectories only, yet deployed on the full Unitree Go1 model in MuJoCo. No plot compares SRB predictions with the full simulator state. The good tracking results suggest the gap is manageable, but it is not shown.

Fourth, the MPC horizon is 6 ms, which is barely more than a one-step lookahead. The gait transitions are fixed-time FSM switches, not optimizer-driven. These facts lower the significance.

The citation pattern is fair. This is a legitimate application of existing machinery, not a breakthrough. But it deserves a serious referee: the hybrid per-phase Koopman idea for legged systems is worth publishing with revisions. I would ask for an input-coupling test, a baseline comparison, an SRB-versus-sim validation plot, and an honest softening of the 'retaining nonlinearity' language.

Bottom line: send it to review, but expect heavy revision.

Referee Report

5 major / 6 minor

Summary. The paper proposes building finite-dimensional linear Koopman predictors for a 2D single rigid body (SRB) model of a quadruped, using Extended Dynamic Mode Decomposition (EDMD) with a physics-informed dictionary, and then using these predictors inside a linear model predictive controller (LMPC) to generate trotting, bounding, and gait transitions in a MuJoCo simulation of the Unitree Go1. The authors report small held-out fitting errors for the Koopman predictors on SRB rollouts, online velocity tracking for both gaits, rough-terrain disturbance rejection, and hand-scheduled trot-to-bound and bound-to-trot transitions. The central claim is that the lifted linear model retains the nonlinearity of the equations of motion, avoiding the need to linearize the model as in standard LMPC.

Significance. If fully validated, this would be a useful demonstration of data-driven lifted linear models for hybrid legged locomotion, with the practical advantage of a convex quadratic program that solves in about 3 ms. The paper has genuine strengths: the EDMD fit is validated on 50 held-out SRB initial conditions and force profiles; the limitations section is candid, noting the accuracy decay over time and the potential benefit of bilinear Koopman realizations; and the simulation results are presented with clear gait diagrams and a video. However, the significance as currently stated is not fully established. The constant-B lifted model cannot represent the state-dependent R(theta)F input coupling in the SRB, no baseline comparison against standard linearized LMPC is provided, the gait transitions are fixed-time finite-state-machine switches rather than optimizer decisions, and the gap between the SRB training model and the full MuJoCo simulation is not quantified. These are load-bearing issues for the paper's claims, but they are addressable with additional experiments and careful rewriting.

major comments (5)

[II-D/E, Eq. (11)] The central claim that the lifted model 'retains the nonlinearity' is not established for the SRB input coupling. In Eqs. (1)-(2), forces enter the translational dynamics through R(theta)F, a product of a state-dependent rotation and the input, yet the EDMD predictor in Eq. (11) has a constant B multiplying u, and the dictionary in Eq. (9) contains [1, p, theta, pdot, thetadot, R thetadot, ..., R thetadot^p] with no observables that multiply a state function by u. A finite linear map from [Pi(x); u] to Pi(x') cannot represent such a product exactly; least-squares EDMD can only fit an average input gain over the training distribution. The authors implicitly acknowledge this in Section IV when they note that a bilinear Koopman model would better represent the SRB control-affine structure. Because the LMPC drives pitch to roughly +/-0.4 rad on rough terrain (Fig. 6), the missing theta-F coupling is a first-order effect in the claimed operating region. Please add a targeted test: report the EDMD prediction error conditioned on pitch magnitude (e.g., |theta| > 0.2 rad) and compare with a bilinear predictor, or otherwise demonstrate that the fitted constant B is adequate over the actual pitch and input range used by the controller.
[II-F, Eqs. (10)-(14)] Section II-F reports a planning horizon of 6 ms and an update horizon of 5 ms, but the number of steps k in Eqs. (10)-(14) and the discretization interval used by the MPC are never stated. The EDMD training data are integrated at dt = 0.001 s, so if the MPC uses the same dt, k would be about 6; if it uses the 5 ms update period, k would be about 1.2. This ambiguity is load-bearing because a horizon of one or two samples is essentially a reactive force law rather than a predictive plan, and it determines whether the claimed 'online optimal control' is meaningful. Please specify k, the MPC discretization time, and the relationship between the 6 ms planning horizon and the 5 ms update period, and report the horizon length used in the experiments.
[III-D, Abstract] Section III-D describes the gait transitions as fixed-time switches between two finite state machines, with the switch from bound to trot placed manually in the flight phase. The Abstract's claim of 'online generation of ... gait transitions' is therefore overstated: the LMPC does not decide or optimize the transition. To support the stated novelty, either add an experiment in which the transition time or phase is chosen by the optimization, or revise the claim to describe hand-scheduled FSM switching with the Koopman LMPC operating within each gait.
[III (all results)] No baseline comparison is provided. The Introduction motivates the work by arguing that linearizing the equations of motion 'may lead to poor solution quality,' but the paper never compares the Koopman LMPC against the standard linearized-SRB LMPC (e.g., ref. [5]) or against a directly linearized model on the same velocity-tracking and rough-terrain tasks. Without such a comparison, the incremental benefit of the Koopman lifting over classical LMPC is not demonstrated, and the reported RMSE values (0.029 for trotting, 0.189 for bounding) have no reference point. Please add a baseline under identical conditions.
[III-A, II-F] The Koopman models are trained exclusively on SRB rollouts (Section III-A), but all locomotion results are obtained in a full MuJoCo simulation of the Unitree Go1. There is no direct comparison between the SRB predictions and the full simulation, so discrepancies due to leg swing, contact modeling, and torso-relative foot positions are invisible to the controller. The small fitting errors in Fig. 3 validate the Koopman approximation of the SRB, not the fidelity of the SRB to the simulated robot. Please report, at minimum, a comparison of SRB and MuJoCo state trajectories under the same planned forces, or quantify the model-mismatch effect on the LMPC tracking error.

minor comments (6)

[Eq. (10)] In the MPC cost (10), the state error uses x_{t+i} but the reference is written x^d_t without a step index; clarify whether the reference is held constant over the horizon or is time-varying.
[Eq. (16)] Equation (16) uses the same symbol c for the lower and upper bounds; use distinct symbols (e.g., c_l and c_u) to avoid ambiguity.
[III-C] Section III-C begins with 'To check the trotting gait reference tracking ability,' but the section is about bounding; this appears to be a copy-editing error.
[II-B, Abstract] Some terminology is nonstandard: 'front foots' and 'rear foots' in Section II-B should be 'front feet' and 'rear feet,' and 'gaits transitions' in the Abstract should be 'gait transitions.'
[III-A, Fig. 3] Figures 3(a)-(d) report fitting errors for the trotting and rear-stance bounding models, but not for the front-stance bounding model, which is also used by the controller (Section II-B); please clarify whether the front-stance model has similar accuracy or was omitted for brevity.
[General] The paper does not state whether the code or the learned Koopman matrices will be released; providing the data-generation script and the learned A and B matrices would improve reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Koopman model is a data-driven EDMD fit validated on SRB rollouts, and the central controller claim is tested in full MuJoCo simulation external to the training data.

full rationale

The paper's derivation chain is: SRB equations (Eqs. 1-2) generate training trajectories; EDMD (Eqs. 5-8) fits a finite-dimensional linear operator (A,B) in a lifted observable space; the LMPC (Eqs. 10-16) uses that operator online; and the controller is evaluated in a full Unitree Go1 MuJoCo simulation. Each link is either an openly acknowledged least-squares fit or an external empirical test, not a definitional equivalence. The validation in Fig. 3 is a roll-out comparison against the same SRB model used to generate training data, which is honest validation of the fit rather than a circular derivation of the SRB from the Koopman model. The MuJoCo trotting, bounding, and gait-transition results are the actual target claims and are external to the EDMD training data. The only self-citation, [23] (Bhounsule and Yang), appears in the swing controller for an analytical inverse kinematics routine and is not load-bearing: it does not justify the Koopman model, the dictionary, or the LMPC. The skeptical concern about the constant input matrix B versus the SRB's state-dependent R(θ)F coupling is a modeling-accuracy or correctness issue, not circularity; the paper itself concedes in Section IV that a bilinear Koopman model would better capture the control-affine structure, showing the limitation is acknowledged rather than hidden. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central demonstration rests on a small number of user-chosen parameters (dictionary order, LMPC weights, gait timings, and MPC horizon) and on domain assumptions about model fidelity. None of these are accompanied by sensitivity studies, and no new physical entities are introduced.

free parameters (4)

Koopman dictionary order p = 4
Number of R*theta^p observables chosen by hand, no sensitivity analysis provided; directly affects the lifted model dimension and accuracy.
LMPC weight matrices Q_i, R_i = not reported
User-chosen diagonal positive definite weights in Eq. (10); influence tracking performance and control effort but no values are given.
Gait timing parameters = trot stance 0.2-0.5s; bound flight/front stance/rear stance 0.1/0.1/0.05s
Hand-specified FSM timings in Section II-F; the gait schedule is fixed, not optimized, and affects transition quality.
MPC planning horizon = 6 ms (planning frequency 166.67 Hz), update 5 ms
Chosen horizon length; the paper does not report the number of steps k or the discretization step, and the very short horizon weakens the claim of nonlinearity retention.

assumptions (4)

domain assumption The Single Rigid Body model, Eqs. (1)-(2), is a valid representation of the quadruped torso dynamics during trotting, bounding, and transitions.
The Koopman model is trained on SRB trajectories, so the controller inherits any SRB modeling error when applied to the full MuJoCo simulation.
domain assumption The chosen dictionary of observables yields a sufficiently accurate finite-dimensional Koopman operator over the operating envelope.
No convergence analysis or dictionary sensitivity study is provided; p=4 is fixed by hand.
domain assumption The finite state machine and swing/stance leg controllers track desired foot positions and forces well enough that torso-level MPC forces are realized.
The paper relies on PD swing control and Jacobian-transpose stance control without quantifying tracking error.
domain assumption MuJoCo simulation is an adequate stand-in for physical hardware for the claimed demonstrations.
Results are simulation-only; no hardware experiments or sim-to-real analysis is provided.

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Cite this review

Pith. "Pith review of Koopman Operator Based Linear Model Predictive Control for 2D Quadruped Trotting, Bounding, and Gait Transition." pith.science (2026). https://pith.science/paper/MPPKOL56

@misc{pith2026250714605,
  author       = {Pith},
  title        = {Pith review of: Koopman Operator Based Linear Model Predictive Control for 2D Quadruped Trotting, Bounding, and Gait Transition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPPKOL56}},
  note         = {Machine review of arXiv:2507.14605}
}

read the original abstract

Online optimal control of quadrupedal robots would enable them to plan their movement in novel scenarios. Linear Model Predictive Control (LMPC) has emerged as a practical approach for real-time control. In LMPC, an optimization problem with a quadratic cost and linear constraints is formulated over a finite horizon and solved on the fly. However, LMPC relies on linearizing the equations of motion (EOM), which may lead to poor solution quality. In this paper, we use Koopman operator theory and the Extended Dynamic Mode Decomposition (EDMD) to create a linear model of the system in high dimensional space, thus retaining the nonlinearity of the EOM. We model the aerial phase and ground contact phases using different linear models. Then, using LMPC, we demonstrate bounding, trotting, and bound-to-trot and trot-to-bound gait transitions in level and rough terrains. The main novelty is the use of Koopman operator theory to create hybrid models of a quadrupedal system and demonstrate the online generation of multiple gaits and gaits transitions.

Figures

Figures reproduced from arXiv: 2507.14605 by the authors.

**Figure 2.** 2D Single Rigid Body model The original method was conceptualized for an uncontrolled system in 1931 by Koopman [14], but only recently tools have been devised to model controlled systems [15]. The applications of the Koopman operator are currently limited to a few simple smooth systems, such as quadcopters [16], underwater vehicles [17], autonomous cars [18], two-link planar manipulators [19], and soft robot manipu… view at source ↗

**Figure 3.** 2D Single Rigid Body model Koopman operator fitting result [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Bounding forward velocity tracking the robot to pitch forward at a small angle to improve forward tracking performance. When the commanded forward velocity is set to zero, the pitch also returns to its neutral position. The trotting gait pattern diagram is shown in […

**Figure 6.** Figure 6: Trotting (top two) and bounding (bottom two) on rough terrain [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: Bounding transit to trotting In this work, we implement simple gaits. Since most gaits are inherently periodic, the gait frequency can be easily modified by changing the cycle duration [7], while gait speed can be adjusted by altering the phase combination and frequenc…

Discussion (0). Continue with ORCID to comment.

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Pith tools

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

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2018

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Trotting, pacing and bounding by a quadruped robot,

M. H. Raibert, “Trotting, pacing and bounding by a quadruped robot,” Journal of biomechanics , vol. 23, pp. 79–98, 1990

work page 1990

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Strogatz, Nonlinear dynamics and chaos

S. Strogatz, Nonlinear dynamics and chaos . Addison-Wesley Read- ing, 1994

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Quadruped bounding con- trol with variable duty cycle via vertical impulse scaling,

H.-W. Park, M. Y . Chuah, and S. Kim, “Quadruped bounding con- trol with variable duty cycle via vertical impulse scaling,” in 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems . IEEE, 2014, pp. 3245–3252

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Model hierarchy predictive control of robotic systems,

H. Li, R. J. Frei, and P. M. Wensing, “Model hierarchy predictive control of robotic systems,” IEEE Robotics and Automation Letters , vol. 6, no. 2, pp. 3373–3380, 2021

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Real-time model predictive control for versatile dynamic motions in quadrupedal robots,

Y . Ding, A. Pandala, and H.-W. Park, “Real-time model predictive control for versatile dynamic motions in quadrupedal robots,” in 2019 International Conference on Robotics and Automation (ICRA) . IEEE, 2019, pp. 8484–8490

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Gait and trajectory optimization for legged systems through phase-based end- effector parameterization,

A. W. Winkler, C. D. Bellicoso, M. Hutter, and J. Buchli, “Gait and trajectory optimization for legged systems through phase-based end- effector parameterization,” IEEE Robotics and Automation Letters , vol. 3, no. 3, pp. 1560–1567, 2018

2018

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Learning torque control for quadrupedal locomotion,

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

2023