{"id":"147c2557-2de6-4ce5-b8e7-b02118ac67ea","arxiv_id":"2501.18220","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"An iterative planner-controller with Gaussian process error learning lets a Pendubot complete swing-ups and unstable transfers in two or three trials despite large model uncertainty.","lead":"The paper presents an iterative method that learns how model error perturbs the active and passive joints of an underactuated robot, then uses that learned model to re-plan and re-control swing-up motions. It shows on a Pendubot that two or three learning iterations are enough to carry the robot to a stabilizing controller's region, even with 30% errors in the dynamic model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Planner cancellation assumption δa=0 in Eq. (7) is unproven during the transient and creates a training/query mismatch for εp; the paper has no bound or repeated-trial evidence for the few-iteration convergence claim.","rationale":"The paper is internally consistent and gives real empirical support: simulations and hardware experiments on a Pendubot, a fair baseline comparison to [7], and a clear description of the learning loop. The reader's conditional verdict is appropriate. My stress-test confirms that the weakest load-bearing premise is the planner's δa = 0 assumption in Eq. (7). What makes it concrete is the induced distribution mismatch for εp: the passive regressor is trained at actual ̈qa values, but during planning it is queried at the idealized ̈qa = u_ref. Since δa is not yet cancelled early in the learning process, these inputs differ, so the planned trajectory's passive dynamics are computed from extrapolated GP predictions. The paper offers no formal bound, no Monte Carlo evidence, and only single-run traces. The self-stated future work on guaranteeing hard constraints during the transient reinforces this gap. This is not a reason to reject: the reported runs are credible and the method is reproducible in principle, but the headline claim of very-few-iteration convergence is not yet supported as a general statement. The verdict should remain conditional, with the condition being a quantitative characterization of the transient cancellation error and its effect on the planned trajectory's feasibility.","tokens_in":10843,"tokens_out":4298,"duration_ms":42698,"concrete_test":"Run scenario 1 with, say, 50 independent random parameter perturbations drawn uniformly in the stated ±30% ranges, recording per-iteration q2 terminal error and LQR-basin entry success. In the same runs, log the actual ̈qa − u_ref during iteration 1 and evaluate εp at the planned query points, comparing with GP training inputs. If any trials fail to reach the basin within four iterations, or if the query/training input gap exceeds the GP length scale early on, the few-iteration convergence claim is not generic and the δa = 0 planning assumption is the limiting factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that after two or three iterations the Pendubot reliably reaches the LQR basin under 30% parameter errors — rests on the planner's prediction model (7)-(8), where δa is set to zero and δp is replaced by εp. This is justified only by the assertion that the control term εa in (9) will cancel δa. During the first iterations the cancellation is imperfect, and the paper provides no bound on ‖δa − εa‖ or on the resulting tracking error. The mismatch is not merely formal: εp is trained on data whose third input is the actual acceleration ̈qa (Sec. III-E), while the planner queries εp at ̈qa = u_ref (Eq. 8). Early on these arguments differ by δa − εa plus PD feedback, so the GP is queried out of distribution. Table I even shows q2 tracking worsening from 0.191 rad (no learning) to 0.623 rad (iteration 1) in experiment scenario 1, before improving at iteration 2; the transient is non-monotonic and uncharacterized. The conclusions list guaranteeing the learning transient as future work, confirming that no such guarantee is claimed. This does not contradict the evidence that the method worked on the reported runs, but it makes 'typically' and 'convergence' unsupported beyond those specific trials.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an iterative learning framework for planning and control of underactuated robots with uncertain dynamics. Each iteration alternates an off-line optimal control phase, which uses a nominal model corrected by a learned perturbation on the passive subsystem, with an on-line partial-feedback-linearization tracking phase, which uses a learned correction for the active subsystem. The perturbations are estimated from measured trajectories via two Gaussian-process regressors fed by position, velocity, and acceleration data. The method is evaluated on a Pendubot in simulation and on hardware for swing-up maneuvers and transfers between unstable equilibria, under deliberate 30% errors in masses and center-of-mass distances, and it is compared against a passivity-based swing-up controller. The central claim is that after two or three iterations the system enters the LQR basin of attraction and completes the maneuver, and that convergence is achieved in very few iterations despite large model uncertainty.","tokens_in":11223,"tokens_out":3079,"duration_ms":33658,"significance":"If the empirical claims hold, the paper makes a useful contribution: it combines learning with optimization-based planning in a way that does not require torque measurements, is not tailored to a specific underactuated mechanism, and explicitly handles state and input constraints. The derivation of the perturbed active/passive dynamics (Eqs. 5-6) and the data-collection formulas (Eqs. 11-12) are clean and well motivated. The inclusion of both simulations and physical experiments, plus a comparison with a classical model-based method under the same uncertainty, strengthens the demonstration. The main weakness is that the paper's broad convergence claims are supported only by a small number of single-run trials and by an unquantified cancellation assumption in the planner; there is no formal convergence result or statistical repeatability evidence. The method appears promising, but the claim 'convergence is reached ... requiring very few iterations' is stronger than what the presented evidence establishes.","major_comments":[{"comment":"The planning phase sets δ_a = 0 in Eq. (7) and justifies this by the claim that the controller will cancel δ_a via ε_a in Eq. (9). This assumption is load-bearing because the planned trajectory, and hence the feasibility of the transfer and the terminal LQR-basin condition, depends on cancellation being accurate already during the early iterations. The paper provides no bound on ‖δ_a − ε_a‖, no characterization of the transient, and no convergence proof; the Conclusions explicitly defer guarantees on the learning transient to future work. As written, 'convergence is reached' in Section IV-A is an empirical statement about the tested runs, not a property of the algorithm. The authors should either provide a formal or probabilistic guarantee under stated assumptions, or substantially weaken the convergence claims in the abstract and introductory bullets.","section":"§III-A, Eq. (7)"},{"comment":"There is a training/query distribution mismatch for the passive regressor ε_p. The training data in Eq. (12) use the actual acceleration q̈_a as an input, while the planner in Eq. (8) queries ε_p at q̈_a = u_ref. Before convergence these two arguments differ by δ_a − ε_a plus the PD feedback terms, so the Gaussian process is queried out of distribution during exactly the iterations where the planned trajectory is most likely to be dynamically infeasible. The non-monotonic transient in Table I (scenario 1, q2 RMSE of 0.191 rad without learning, 0.623 rad at iteration 1, then 0.038 rad at iteration 2) indicates that the mismatch can significantly degrade performance before it improves. The paper should either quantify the mismatch and its effect on planning, or present this as a known limitation rather than as a step in a monotone convergence process.","section":"§III-E and §III-A, Eqs. (8) and (12)"},{"comment":"The experimental evaluation consists of single runs per scenario and per iteration, with no repeated trials, error bars, or confidence statistics. Given that the active regressor is updated online and that the passive regressor depends on filtered numerical differentiation, run-to-run variability could be substantial. Consequently, the abstract's 'typically' and the bullet 'convergence is reached ... requiring very few iterations' are not supported as statistical statements. The authors should either repeat each experiment several times and report mean/variability, or restrict the claims to 'in the reported trials, convergence was observed after two or three iterations.'","section":"§IV-B and Table I"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'V ery' should be 'Very'.","section":"Abstract"},{"comment":"The sentence 'Each iteration includes an off-line planning phase and an on-line planning phase' should likely read 'an on-line control phase', consistent with Section III; as written, the sentence describes two planning phases.","section":"§V, Conclusions"},{"comment":"The statement that the separate use of one regressor in planning and the other in control 'proves to be computationally more efficient' is not quantified; a brief explanation (e.g., number of optimizer evaluations or real-time constraints) would help the reader assess the claim.","section":"§III-A, second paragraph"},{"comment":"The 3R Pendubot simulation is only referenced through the accompanying video; providing a quantitative result (e.g., tracking error or number of iterations to convergence) would make the higher-underactuation evidence more verifiable.","section":"§IV-A, paragraph after Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The paper reports a well-executed empirical study with clean derivations and a useful hardware demonstration. The main issue is that the title and abstract claim convergence properties that the paper does not prove and that the single-run experiments do not statistically establish. I believe this is fixable within the scope of the manuscript: either add repeated trials and report variability, or soften the convergence language to describe demonstrated behavior. A referee should also consider whether the lack of a convergence proof is acceptable for the venue; if the authors choose to keep the strong claims, the proof would be required. The self-citation to [22] is appropriate and not excessive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a legitimate and clearly presented extension of the authors' earlier learning-based feedback linearization work to underactuated robots. The split-regressor design—epsilon_p correcting the planner, epsilon_a correcting the controller—is genuinely new relative to their prior fully actuated method and to the closest related work. The empirical claims are supported for the reported runs: the Pendubot reaches the LQR basin after two or three iterations under 30% parameter errors, and the comparisons to the model-based baseline [7] are fair—that method works with the exact model and fails under the same uncertainty. The perturbed dynamics derivation and the data collection formulas are clean, the two experimental scenarios are well chosen, and the simulated 3R case is a useful stress test. Credit where due: the paper does what it claims, honestly, and the writing is refreshingly direct about what was and was not done.\n\nThe soft spots are real but not disqualifying. The stress-test note about delta_a = 0 in the planning model (Eq. 7) has genuine teeth. The planner assumes the active perturbation is canceled while learning is still converging, and epsilon_p is trained on data that include the actual acceleration while the planner queries it at u_ref. Early on those arguments can differ by delta_a minus epsilon_a plus PD feedback, so the GP is being queried out of distribution. Table I confirms the transient is non-monotonic: in scenario 1, q2 tracking error goes from 0.191 rad without learning to 0.623 rad at iteration 1 before dropping to 0.038 at iteration 2. The paper provides no bound on that transient, and the authors explicitly list guaranteeing the learning transient as future work. So the concern is not that the paper overclaims—it doesn't—but that the method's convergence behavior is empirically demonstrated rather than formally understood. For a paper advertising 'very few iterations typically,' I would have liked repeated trials and some sensitivity analysis, since swing-up success near a basin boundary can be threshold-sensitive.\n\nThe citation pattern is fine; the self-citation to [22] is appropriate because this is a direct extension. No code or data release makes full reproduction harder, but the video and the detailed experimental section help.\n\nThis is worth serious refereeing. I would send it out, and I would cite it when discussing learning-based planning and control for underactuated systems. My verdict is conditional rather than unconditional: the authors should address the transient assumption or explicitly scope the claims to the demonstrated runs, and provide at least a small number of repeated trials.","headline":"A solid, honest extension of the authors' earlier learning-based feedback linearization work to underactuated robots, with real Pendubot experiments; the main gaps are a missing transient guarantee and thin statistics, not a fatal flaw.","tokens_in":760,"tokens_out":1079,"would_cite":true,"duration_ms":31132,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A two-regressor learning loop lets an underactuated Pendubot swing up in two or three iterations despite 30 percent model errors.","keywords":["underactuated robots","model uncertainty","Gaussian process regression","partial feedback linearization","iterative learning control","trajectory planning","Pendubot","swing-up control"],"falsifier":"Repeat the swing-up simulation with the same 30 percent parameter errors but set the active-side learned correction to zero for the first two iterations while keeping the passive-side learning active; if the maneuver still converges, the active correction is not necessary, and if it fails, the paper's reliance on active perturbation cancellation is confirmed.","tokens_in":10650,"feed_emoji":"🤖","tokens_out":6215,"duration_ms":51123,"temperature":0.7,"pith_summary":"The paper proposes an iterative method that lets an underactuated robot plan and execute a transfer between equilibria when its dynamic model is badly wrong. Each iteration alternates an off-line optimization that plans a trajectory using a learned correction for the passive joints with an on-line controller that tracks that trajectory using a learned correction for the active joints. The claim is that this two-sided learning makes the planned trajectory dynamically feasible and the tracking accurate enough that, after two or three iterations, the Pendubot reaches the basin of attraction of a balancing LQR and completes the swing-up. If true, the approach would remove the usual requirement for an accurate model in underactuated motion planning and control, and would apply to any underactuated robot and any equilibrium-transfer maneuver.","feed_headline":"In two or three tries, a two-regressor learner swings up a Pendubot","feed_subtitle":"Even with model masses off by 30 percent, the robot reaches balance control in two or three attempts.","key_machinery":"The load-bearing mechanism is the pair of perturbation equations that result from applying collocated partial feedback linearization to the nominal model. They express the actual accelerations of the active and passive coordinates as the nominal commanded acceleration plus unknown perturbation terms. Each iteration of the algorithm solves an optimal control problem over a prediction model that sets the active perturbation to zero and replaces the passive perturbation with its current estimate, then runs a tracking controller whose commanded active acceleration is the planned feedforward plus PD feedback minus the learned active correction. Gaussian process regressors with squared-exponential kernels are used, one updated in real time for the active term and one updated at the end of each trial for the passive term, the latter reusing all data from previous trials.","core_discovery":"The central discovery is that model uncertainty in an underactuated robot can be handled by learning two separate perturbation terms rather than by correcting the whole model. After a collocated partial-feedback-linearization control is applied on the nominal model, the perturbed closed-loop dynamics split into an active subsystem, where the uncertainty appears as a term added to the commanded acceleration, and a passive subsystem, where it appears as a term added to the nominal passive acceleration. The paper learns Gaussian-process estimates of these two terms, uses the passive estimate in the planner's prediction model and the active estimate in the tracking controller, and reruns the planning-control loop with warm starts. In simulations with link masses increased by 30 percent and center-of-mass distances reduced by 30 percent, the Pendubot performs a swing-up from the stable down-down equilibrium to the unstable up-up equilibrium after three iterations, and a transfer between two unstable equilibria after two iterations; experiments complete the swing-up in two iterations. The paper states that the same scheme works for a three-link Pendubot with two passive joints and requires no torque measurements.","pith_inferences":["A testable extension would be to drive the planner with the full Gaussian-process posterior, using the predictive variance of the passive perturbation in the optimization, which could enforce hard state constraints during the learning transient while still converging in few iterations.","The separation of roles, passive correction in planning and active correction in tracking, suggests a general principle for iterative learning on underactuated systems: plan with the term that shapes the passive dynamics and track with the term that the controller can directly cancel.","If the method generalizes as claimed, similar loops could be applied to other underactuated platforms such as quadrotors or humanoids, where model uncertainty is often significant and planning under dynamic constraints is hard; the authors list these as future targets.","A stress test beyond the paper's 30 percent perturbation, for instance 50 or 60 percent parameter errors, would show whether the iteration count grows gracefully or whether a feasibility barrier appears; the paper does not report such a sweep."],"forward_implications":["If the paper is right, a Pendubot can reliably swing up to an inverted equilibrium after two or three learning iterations even when the nominal model overestimates link masses by 30 percent and underestimates center-of-mass distances by 30 percent.","The same two-regressor loop, without modification, handles transfers between unstable equilibria and higher underactuation, such as a 3R Pendubot with two passive joints, suggesting it is not specialized to one maneuver.","Because neither torque measurement nor exact model knowledge is required, the method can be implemented from encoder measurements alone, broadening its applicability to robots without joint torque sensors.","Constraints on states and inputs can be folded into the planning optimization, so the approach can be extended to tasks with obstacles or actuation limits in later iterations.","The paper's comparison indicates that a purely model-based passivity swing-up controller that succeeds with the exact model fails under the same uncertainty, so the learned corrections are what account for the success."],"supporting_citations":[{"why":"Provides the partitioned dynamic model for underactuated manipulators that the perturbation split is built on.","marker":"[3]"},{"why":"Introduces partial feedback linearization, the control structure used to obtain the perturbed closed-loop dynamics.","marker":"[4]"},{"why":"The passivity-based swing-up controller used as the comparison baseline that fails under the same model uncertainty.","marker":"[7]"},{"why":"The authors' earlier on-line learning procedure for feedback linearization of fully actuated robots, which this paper extends to underactuated systems.","marker":"[22]"},{"why":"Supplies the Gaussian process regression framework used to estimate the perturbation terms.","marker":"[29]"},{"why":"Provides the information-gain sparse selection criterion that keeps the active-side regressor cheap enough for real-time use.","marker":"[30]"},{"why":"Gives the Pendubot dynamic model and nominal parameter values used in both simulations and experiments.","marker":"[31]"}],"fun_headline_variants":["Two tries to swing-up: learning beats 30% model error","Learning two perturbation terms swings up a Pendubot in 2-3 tries","Pendubot swing-up with 30% wrong masses: two tries suffice","Two-iteration swing-up: Gaussian processes handle 30% model error","Underactuated? Learn two effects, swing up in two tries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The planner assumes that the learned active-joint correction will cancel the active perturbation during the control phase, even in the early iterations before the regressor has converged, and the paper provides no formal bound on the error during that transient.","fun_headline_variants_meta":{"raw":{"variants":["Two tries to swing-up: learning beats 30% model error","Learning two perturbation terms swings up a Pendubot in 2-3 tries","Pendubot swing-up with 30% wrong masses: two tries suffice","Two-iteration swing-up: Gaussian processes handle 30% model error","Underactuated? Learn two effects, swing up in two tries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000495,"raw_usage":{"total_tokens":2401,"prompt_tokens":893,"completion_tokens":1508,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":1409}},"tokens_in":509,"tokens_out":1508,"duration_ms":11058,"temperature":1.0,"reasoning_tokens":1409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:15:28.719654+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the swing-up simulation with the same 30 percent parameter errors but set the active-side learned correction to zero for the first two iterations while keeping the passive-side learning active; if the maneuver still converges, the active correction is not necessary, and if it fails, the paper's reliance on active perturbation cancellation is confirmed.","supporting_citations":[{"cited_title":"Underactuated manipulators: Control properties and techniques,","cited_arxiv_id":null,"evidence_quote":"Provides the partitioned dynamic model for underactuated manipulators that the perturbation split is built on."},{"cited_title":"Partial feedback linearization of underactuated mechanical systems,","cited_arxiv_id":null,"evidence_quote":"Introduces partial feedback linearization, the control structure used to obtain the perturbed closed-loop dynamics."},{"cited_title":"Partial stabilization of underactu- ated Euler–Lagrange systems via a class of feedback transformations,","cited_arxiv_id":null,"evidence_quote":"The passivity-based swing-up controller used as the comparison baseline that fails under the same model uncertainty."},{"cited_title":"An online learning procedure for feedback linearization control without torque measurements,","cited_arxiv_id":null,"evidence_quote":"The authors' earlier on-line learning procedure for feedback linearization of fully actuated robots, which this paper extends to underactuated systems."},{"cited_title":"Rasmussen and C","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian process regression framework used to estimate the perturbation terms."},{"cited_title":"Fast forward selection to speed up sparse Gaussian process regression,","cited_arxiv_id":null,"evidence_quote":"Provides the information-gain sparse selection criterion that keeps the active-side regressor cheap enough for real-time use."},{"cited_title":"Enforcing constraints over learned policies via nonlinear MPC: Application to the Pendubot,","cited_arxiv_id":null,"evidence_quote":"Gives the Pendubot dynamic model and nominal parameter values used in both simulations and experiments."}],"review_version":1}