{"id":"1b3e5fb4-e063-465e-be5a-045f5ecc2685","arxiv_id":"2507.14605","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Koopman-EDMD linear models of a 2D single rigid body enable online linear MPC of trotting, bounding, and gait transitions in a simulated quadruped.","lead":"Researchers used Koopman operator theory to create a linear model of a simulated four-legged robot's body, then ran it inside a fast model predictive controller to make the robot trot, bound, and switch gaits, including on rough terrain. The work shows a data-driven path to real-time optimization-based legged control.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Constant B in the lifted linear model (Eq. 11) cannot represent the state-dependent input coupling R(θ)F in Eq. (1), so the 'retaining nonlinearity' claim is not established; a targeted test at nonzero pitch is needed.","rationale":"The central claim has two parts: (i) Koopman-EDMD produces linear models that retain SRB nonlinearity, and (ii) those models support online LMPC for multiple gaits and transitions in MuJoCo. Part (ii) is supported by the simulations, assuming they were run as reported. Part (i) is where the argument is least secure. The control-affine structure of the SRB means the true lifted system is bilinear in state and input (Bruder et al., cited [26]); a constant-B linear model is a restricted approximation. The paper's own Section IV concedes this. The model-fit experiment in Fig. 3 uses random initial conditions and force profiles but does not report the ranges; if pitch excursions are small, the fit cannot reveal the missing θF term. The rough-terrain results show pitch up to about 0.4 rad, so the missing coupling is relevant in the claimed operating regime. The proposed one-step test is cheap and decisive: if the error grows with |θF|, the 'retaining nonlinearity' claim should be downgraded to 'a short-horizon linearized model with a physics-informed lift.' The SRB-to-MuJoCo concern raised by the reader remains valid but is secondary: even a perfect SRB representation would not rescue the model if the dictionary cannot encode the input-state product. Thus the verdict stays conditional, with an added required test.","tokens_in":9638,"tokens_out":12095,"duration_ms":155781,"concrete_test":"Evaluate one-step prediction error of the trained Koopman model on a hold-out SRB test set with θ fixed at values in {-0.4, -0.2, 0, 0.2, 0.4} rad and F varying over the controller's full range. If the ṗ error scales with |θF|, the constant-B model is not capturing R(θ)F. As a second check, retrain EDMD with a dictionary augmented by the four entries of R (cosθ and sinθ) and compare hold-out error; if the augmented model is materially better, the paper's dictionary is the limiting factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the SRB model (Eqs. 1-2), the control forces F enter the translational dynamics through R(θ)F, a product of a state-dependent rotation and the input. The Koopman predictor (Eqs. 4 and 11) has a constant B multiplying u, and the dictionary (Eq. 9) contains [1, p, θ, ṗ, θ̇, Rθ̇, ..., Rθ̇^p] but no feature that multiplies an input by a state function. A finite linear map from [Π(x); u] to Π(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, where they note that a bilinear Koopman model would better represent the SRB control-affine structure [26]. Consequently, the small fitting errors in Fig. 3 do not by themselves show that nonlinearity is retained; they may reflect a narrow training range. Since the LMPC can drive pitch to roughly ±0.4 rad on rough terrain (Fig. 6), the missing θF coupling is a first-order effect for the claimed operating region.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9850,"tokens_out":6748,"duration_ms":77888,"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":[{"comment":"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.","section":"II-D/E, Eq. (11)"},{"comment":"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.","section":"II-F, Eqs. (10)-(14)"},{"comment":"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.","section":"III-D, Abstract"},{"comment":"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.","section":"III (all results)"},{"comment":"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.","section":"III-A, II-F"}],"minor_comments":[{"comment":"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.","section":"Eq. (10)"},{"comment":"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.","section":"Eq. (16)"},{"comment":"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.","section":"III-C"},{"comment":"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.'","section":"II-B, Abstract"},{"comment":"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.","section":"III-A, Fig. 3"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a simulation-only demonstration, and the authors' claims are currently broader than what is shown: the nonlinearity-retention claim, the predictive nature of the 6 ms horizon, and the online generation of transitions all need to be reined in or supported with targeted experiments. The authors are candid about limitations in Section IV, which is appreciated. If the revised paper adds a baseline comparison, a pitch-conditioned validation of the Koopman predictor, and a clarification of the MPC horizon and transition mechanism, it could be a useful contribution to the legged-locomotion MPC literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2507.14605.\n\nThe 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.\n\nNow 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.\n\nSecond, 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.\n\nThird, 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.\n\nFourth, 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.\n\nThe 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.\n\nBottom line: send it to review, but expect heavy revision.","headline":"Koopman-EDMD LMPC for a simulated quadruped: a real application, but the 'retains nonlinearity' claim does not survive the constant-B input coupling.","tokens_in":10417,"tokens_out":2485,"would_cite":false,"duration_ms":29516,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Koopman operator","extended dynamic mode decomposition","linear model predictive control","quadruped locomotion","gait transition","trotting","bounding","hybrid systems"],"falsifier":"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.","tokens_in":9389,"feed_emoji":"🤖","tokens_out":13357,"duration_ms":126609,"temperature":0.7,"pith_summary":"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.","feed_headline":"A single Koopman linear model drives trot, bound, and back","feed_subtitle":"Nonlinear dynamics lifted into a linear space let one controller trot, bound, and switch.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Koopman operator whose finite approximation is the control model used throughout.","marker":"[14]"},{"why":"Provides the EDMD least-squares regression that turns the roll-out data into the matrices A and B.","marker":"[15]"},{"why":"Supplies the single-rigid-body model and convex LMPC formulation that this work lifts into Koopman space.","marker":"[5]"},{"why":"Source of the physics-informed observable dictionaries used to build the lifted state.","marker":"[16]"},{"why":"The MuJoCo simulator in which all trotting, bounding, and transition experiments are run.","marker":"[22]"},{"why":"Stance-phase planning inspiration and the flight-phase switch rule used for the bound-to-trot transition.","marker":"[1]"},{"why":"The qpSWIFT solver that makes the online quadratic program fast enough for real-time use.","marker":"[24]"},{"why":"Provides the analytical inverse kinematics and leg controller that realizes swing and stance commands.","marker":"[23]"}],"fun_headline_variants":["Koopman lifts nonlinearity, one LMPC trots, bounds, switches","Hybrid Koopman models: single LMPC for trot, bound, and transition","Trot, bound, switch: Koopman-based LMPC works on rough terrain","Nonlinear dynamics lifted, online LMPC handles gait changes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Koopman lifts nonlinearity, one LMPC trots, bounds, switches","Hybrid Koopman models: single LMPC for trot, bound, and transition","Trot, bound, switch: Koopman-based LMPC works on rough terrain","Nonlinear dynamics lifted, online LMPC handles gait changes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000276,"raw_usage":{"total_tokens":1648,"prompt_tokens":947,"completion_tokens":701,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":612}},"tokens_in":563,"tokens_out":701,"duration_ms":8639,"temperature":1.0,"reasoning_tokens":612,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:51:41.936837+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hamiltonian systems and transformation in hilbert space,","cited_arxiv_id":null,"evidence_quote":"Defines the Koopman operator whose finite approximation is the control model used throughout."},{"cited_title":"A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Provides the EDMD least-squares regression that turns the roll-out data into the matrices A and B."},{"cited_title":"Se (3) koopman-mpc: Data-driven learning and control of quadrotor uavs,","cited_arxiv_id":null,"evidence_quote":"Source of the physics-informed observable dictionaries used to build the lifted state."},{"cited_title":"Trotting, pacing and bounding by a quadruped robot,","cited_arxiv_id":null,"evidence_quote":"Stance-phase planning inspiration and the flight-phase switch rule used for the bound-to-trot transition."},{"cited_title":"A simple controller for omnidirectional trotting of quadrupedal robots: Command following and waypoint tracking,","cited_arxiv_id":null,"evidence_quote":"Provides the analytical inverse kinematics and leg controller that realizes swing and stance commands."}],"review_version":1}