{"id":"71ecd145-e60f-444c-9cef-ccfa53268ff1","arxiv_id":"2507.23203","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"A centroidal-dynamics MPC controller with thrust-vectoring enables a simulated quadruped to walk on a narrow beam and reject lateral pushes.","lead":"A simulated quadruped robot with knee-mounted thrusters walks along a ten-centimeter-wide beam and recovers from a side push using a model predictive control scheme. The thrusters, normally used for flight, help stabilize the robot's sideways balance and keep its feet from slipping.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (7) lists \\dot\\omega as a state, but the A matrix then makes \\dot\\theta = R_z^\\top \\dot\\omega; since Eq. (5) requires \\dot\\theta = R_z^\\top \\omega and no \\omega state exists, the MPC prediction model is dimensionally inconsistent as written.","rationale":"The reader identified the small-angle approximation in Eq. (5) and simulation fidelity as the weakest assumptions. I agree that those are relevant, but the more load-bearing issue is that the state vector and the A matrix in Eq. (7) are internally inconsistent. The paper defines the third state as \\dot\\omega and then uses that state in the orientation row as though it were \\omega. This is not merely a modeling approximation; it is a dimensional mismatch in the prediction model. Since the MPC's predicted states are the basis for all reported results, the central claim cannot be evaluated from the text alone. The authors may have intended the state to be \\omega, in which case the notation is a typo that also appears in Section IV, but without code or a corrected manuscript we cannot confirm that the reported simulations use a valid model. I would therefore withhold a decision rather than conditionally accept or reject, pending a corrected Eq. (7) and a verification that the simulation results are robust to the state definition change.","tokens_in":7615,"tokens_out":8670,"duration_ms":106867,"concrete_test":"Implement Eq. (7) exactly as printed with initial state \\theta = 0, \\omega = (0.5, 0, 0) rad/s, and \\dot\\omega = 0. The printed model predicts \\dot\\theta = 0, while Eq. (5) and a one-step PyBullet rollout predict \\dot\\theta \\neq 0. Then re-run the beam-walking and 40 N push-recovery simulations with the corrected state vector [\\theta, p, \\omega, \\dot p]^\\top. If the reported behaviors change materially, the missing \\omega-to-\\theta integrator is load-bearing; if they persist unchanged, the \\dot\\omega notation is a typo and the earlier conditional verdict remains appropriate.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"In Section III-A, the state is defined as x = [\\theta^\\top, p^\\top, \\dot\\omega^\\top, \\dot p^\\top]^\\top, and this definition is repeated in Section IV. Equation (5) gives the kinematic relation \\dot\\theta \\approx R_z^\\top \\omega. However, in the linear model (7), the first block row of A multiplies the third state component by R_z^\\top. If that component is \\dot\\omega, then the model predicts \\dot\\theta = R_z^\\top \\dot\\omega, not R_z^\\top \\omega. There is no state variable for \\omega and no integrator from \\dot\\omega to \\omega, so the orientation prediction is driven by angular acceleration instead of angular velocity. This is dimensionally incompatible: a nonzero angular velocity with zero angular acceleration would produce no orientation change in the model. The error is not a small-angle approximation issue; it persists even under the stated small roll and pitch conditions. The subsequent discretization in Eq. (11) inherits the same defect. Because this prediction model is the core of the MPC, the central claim that the controller is based on a valid centroidal dynamics model is not supported by the manuscript as written. If the intended state was \\omega, the notation appears twice and should be corrected; if the state truly includes \\dot\\omega, the model is missing a necessary state and equation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a simulation study of thruster-assisted narrow-beam walking for the Husky Beta quadruped. The authors formulate a centroidal dynamics model with ground reaction forces and propeller thrusts as inputs, linearize the attitude kinematics under a small-angle approximation, and build a model predictive controller that solves a quadratic program at 100 Hz. The paper reports two simulation experiments: a 40 N lateral push-recovery comparison with and without thrusters, and a 10 s walk on a 0.1 m wide beam. The claimed contribution is that a single convex MPC framework can coordinate foot forces and thrust forces to expand the locomotion envelope of a legged-aerial platform in simulation.","tokens_in":7976,"tokens_out":7227,"duration_ms":77948,"significance":"If the central claims held, the paper would provide a useful demonstration of convex MPC with thrust vectoring for multi-modal legged-aerial locomotion, and the push-recovery comparison is a well-posed control experiment. The authors should be credited for formulating the QP with friction-cone constraints, for using a high-fidelity PyBullet model, and for reporting thruster-force and friction-ratio trends. However, the current manuscript contains a state-vector inconsistency in the prediction model, unconstrained swing-foot forces in the QP, a missing no-thruster baseline for beam walking, and unreported controller parameters, so the evidence does not yet support the paper's central claim as written.","major_comments":[{"comment":"The state is defined as x = [θ^T, p^T, ˙ω^T, ˙p^T]^T, but the first block row of A multiplies the third state component by R_z^T. If the third component is ˙ω, the model predicts ˙θ = R_z^T ˙ω, which is dimensionally inconsistent with Eq. (5), ˙θ ≈ R_z^T ω. There is no ω state or integrator, so a nonzero angular velocity with zero angular acceleration would produce no orientation change in the prediction model. Because this prediction model is the core of the MPC, the state definition must be corrected to include ω (or the A matrix, Eq. (7), and the repeated definition in Section IV must be changed accordingly).","section":"Section III-A and IV, Eq. (7)"},{"comment":"The control input u includes u_{g,i} for all four legs and the B matrix in Eq. (7) includes all four ground-force columns, but the friction-cone constraints in Eq. (12) only bound u_{g,i} for i ∈ S_t. For swing legs, no constraint forces u_{g,i} = 0, so the QP can command nonzero ground reaction forces on feet that are not in contact, producing unphysical predicted dynamics and joint torques through Eq. (9). Please add equality constraints u_{g,i} = 0 for swing legs or remove the swing-leg ground-force columns from the prediction model.","section":"Section IV, Eq. (12)"},{"comment":"The beam-walking experiment is only run with thrusters enabled. Because the paper's contribution is the thruster-assisted expansion of the locomotion envelope, a no-thruster baseline is needed to establish that the beam walking is enabled or improved by the thrusters. The push-recovery experiment provides such a baseline, but Section V-C does not, so the claim that the sagittal propeller played a significant role in satisfying the friction cone constraint is not supported by an ablation.","section":"Section V-C"},{"comment":"The reported simulation setup gives only the horizon (5) and the QP update rate (100 Hz). The state and control weights Q and R, friction coefficient μ_s, thruster limit u_max, stance time T_s, discretization step Δt, and PD gains are not reported. Without these values, the results are not reproducible and the reported friction-ratio and thrust-force margins cannot be interpreted quantitatively.","section":"Section V-A and Eq. (12)"},{"comment":"The small-angle linearization of Eq. (5) is load-bearing for the MPC prediction, but the paper does not report the actual roll and pitch excursions during the beam walk or the push recovery. The reader cannot verify that the robot remains in the regime where the approximation is valid. Please report the Euler-angle time histories or numerical maxima, and specify the quantitative bounds implied by the small-angle assumption.","section":"Section III-A and V-C"}],"minor_comments":[{"comment":"The continuous-time gravity term h_g should be multiplied by Δt in the discretized expression, or A_k and B_k should be defined as exact discretization matrices. As written, x_{k+1} = A_k x_k + B_k u_k + h_g mixes continuous and discrete quantities.","section":"Eq. (11)"},{"comment":"The phrase '0n and 1n donates' should read '0_n and 1_n denote'.","section":"After Eq. (7)"},{"comment":"The phrase 'the robot never heads upward' is informal; please state the quantitative small-angle bounds used for the linearization.","section":"Eq. (5)"},{"comment":"The references to 'snapshots 2-5 in Figure 5(a)' and 'snapshots 6-8 in Figure 5(b)' are unclear because the multi-panel figure layout is not described; please label the snapshots and describe the panel contents in the caption.","section":"Section V-B"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThis paper reports a simulation study of a thruster-assisted quadruped walking on a narrow beam, using a centroidal-dynamics MPC. The genuinely new part is the scenario: applying this convex MPC framework to narrow-beam walking with knee-mounted thrusters. The thrust-vectoring idea itself is drawn from the authors' prior Harpy work, so the novelty is incremental. The push-recovery experiment, contrasting the same controller with and without thrusters, is a clean and useful ablation; that part deserves credit.\n\nThe problem is in the core linear model. Equation (7) defines the state as x = [θ, p, \\dotω, \\dotp]^T, but the A matrix puts R_z^T in the first block row multiplying the third block. For the kinematic relation \\dotθ = R_z^T ω to hold, that third state must be ω, not \\dotω. As written, the model predicts orientation from angular acceleration, which is dimensionally inconsistent. This is not a small-angle approximation issue; it is a mismatch in the state-space definition. The same definition appears again in Eq. (11), so it is either a persistent typo or a genuine modeling error. Either way, the prediction model is load-bearing and incorrect as written.\n\nOther soft spots: the beam-walking demonstration has no no-thruster baseline, so we do not know what the thrusters actually buy in that scenario. The cost weights Q, R, the friction coefficient, and the thruster limits are not reported, which makes the simulation hard to reproduce. The small-angle assumption in Eq. (5) is load-bearing but never checked against the simulated attitudes. And there is no hardware validation, which the authors acknowledge as future work.\n\nWhat the paper does well: the formulation is otherwise standard and clear, the push-recovery comparison is a valid control experiment, and the simulations show plausible behavior. The writing is readable and the related work is adequately covered, though heavily self-referential.\n\nI would send this to peer review because the topic is timely and the model error appears fixable. A serious referee should focus on the state-space inconsistency and the missing baselines before the claims can be accepted. This is not a paper I would cite in its current form.","headline":"A standard centroidal-MPC simulation study with a serious state-space inconsistency in its linear model; the push-recovery ablation is solid, but the beam-walking claims need the fix and more baselines.","tokens_in":8545,"tokens_out":5228,"would_cite":false,"duration_ms":56877,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single convex controller lets a thrustered quadruped walk a 0.1 m beam and recover from a 40 N push.","keywords":["quadrupedal locomotion","thruster-assisted walking","model predictive control","centroidal dynamics","narrow-path walking","push recovery","quadratic programming","legged-aerial robot"],"falsifier":"Run the same ten-second beam-walking task and 40-newton push on the physical robot: if its attitude drifts, it leaves the beam, or it tips over while the simulator stayed upright, the sim-to-hardware transfer claim fails. A quicker in-simulation check is to initialize the walk with a roll angle outside the small-angle range, say 15 to 20 degrees, and see whether the linearized MPC still holds the beam; losing stability there would confirm the small-angle assumption as the effective limit.","tokens_in":7436,"feed_emoji":"🤖","tokens_out":5760,"duration_ms":59724,"temperature":0.7,"pith_summary":"This paper argues that a single model-predictive controller, built on centroidal dynamics and solved as a quadratic program, can coordinate four ground-reaction forces with four knee-mounted thruster forces to stabilize a quadruped robot's frontal dynamics. In simulation, this lets the robot walk steadily along a beam only 0.1 meters wide and recover from a 40-newton lateral push applied for half a second. The authors position this as a step toward a robot that can traverse narrow or flexible pathways that ordinary legged locomotion cannot manage. The payoff, if correct, is that posture and thrust control need not be separate: one convex optimization framework handles both.","feed_headline":"Quadruped walks a 0.1 m beam using knee thrusters","feed_subtitle":"In simulation, MPC coordinates feet and propellers, recovering from a 40 N lateral push.","key_machinery":"The load-bearing object is the centroidal dynamics model with augmented thruster inputs: the robot's center-of-mass acceleration and angular acceleration are expressed as linear functions of four ground-reaction force vectors and four scalar thruster forces, with lever arms updated in real time from leg kinematics. Under the assumption of small roll and pitch angles, the attitude kinematics are linearized, yielding a convex model-predictive control problem whose constraints include a linearized friction cone and single-direction thrust limits. The optimizer selects both foot forces and thruster forces at 100 Hz, making the thrusters an active part of the stability loop rather than a separate assist.","core_discovery":"The central claim is that lateral stability on a narrow path can be achieved by adding thruster forces as inputs to a linearized centroidal dynamics model, then solving a short-horizon model-predictive control problem with ground-reaction-force friction-cone constraints and thrust bounds. The simulation shows the robot maintains a stable lateral position and stance height for ten seconds of beam walking, with thruster forces staying below 7 newtons, and recovers from a 40-newton lateral disturbance that causes the no-thruster controller to fall within two seconds. The authors conclude that thrusters effectively enlarge the feasible region of ground contact forces, so the no-slip condition can be met even when foot placement is severely restricted.","pith_inferences":["If this transfers to hardware, the approach implies that trajectories previously classified as infeasible for legged robots due to narrow support areas become feasible once additional unilateral forces are available; the effective support polygon could be widened without widening the foot contact area.","The paper's stated next target, walking on a flexible rope, would require extending the model from a rigid beam to time-varying contact geometry; the same convex MPC structure could still work if the reference states and contact point set are updated online.","A testable extension is to measure the minimum beam width the controller can sustain as a function of maximum thrust; this would quantify how thrust authority trades off against required foot-placement precision.","Because the thrusters act almost directly on the body's lateral dynamics, the framework may also apply to bipedal robots or to recovering from pushes on stairs, not just quadrupeds on beams."],"forward_implications":["The same MPC formulation covers both nominal beam walking and large lateral disturbance rejection, so no separate recovery-mode controller is needed.","Because thruster forces on the beam stay below 7 newtons (about 7 percent of maximum thrust), only modest thrust authority is needed to stabilize roll dynamics in this scenario.","Without thrusters the same controller fails within two seconds under the 40-newton push, showing the thrusters are what expand the recoverable disturbance envelope.","The friction-cone constraints are satisfied for all four feet during beam walking, indicating the thrusters offload lateral ground forces enough to prevent slipping."],"supporting_citations":[{"why":"Supplies the convex model-predictive control formulation for dynamic legged locomotion that this paper adapts to include thruster inputs.","marker":"[29]"},{"why":"Provides the foot-placement heuristic used by the swing-phase controller to set step locations.","marker":"[30]"},{"why":"Supplies the physics simulator in which the beam-walking and push-recovery experiments are run.","marker":"[31]"},{"why":"Provides the QP solver used to solve the MPC optimization at 100 Hz in the simulations.","marker":"[26]"},{"why":"Establishes the original thruster-assisted legged design and control direction that this work extends to narrow-path walking.","marker":"[6]"},{"why":"Documents the solver benchmark that justifies choosing the QP solver for real-time performance.","marker":"[25]"}],"fun_headline_variants":["Knee thrusters keep quadruped on 10 cm beam","Simulated quadruped walks a 0.1 m beam with jets","Thruster-assisted walking: quadruped balances on narrow path","Quadruped uses thrusters to recover from 40 N push","QP + thrusters: stable narrow-path walking in simulation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The controller's prediction model assumes the robot's roll and pitch angles stay small, and the simulation assumes the physical robot's thrusters and ground contacts behave as modeled; if either fails in hardware, the demonstrated stability may not transfer.","fun_headline_variants_meta":{"raw":{"variants":["Knee thrusters keep quadruped on 10 cm beam","Simulated quadruped walks a 0.1 m beam with jets","Thruster-assisted walking: quadruped balances on narrow path","Quadruped uses thrusters to recover from 40 N push","QP + thrusters: stable narrow-path walking in simulation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000161,"raw_usage":{"total_tokens":1196,"prompt_tokens":869,"completion_tokens":327,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":239}},"tokens_in":485,"tokens_out":327,"duration_ms":4614,"temperature":1.0,"reasoning_tokens":239,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:56:40.684094+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same ten-second beam-walking task and 40-newton push on the physical robot: if its attitude drifts, it leaves the beam, or it tips over while the simulator stayed upright, the sim-to-hardware transfer claim fails. A quicker in-simulation check is to initialize the walk with a roll angle outside the small-angle range, say 15 to 20 degrees, and see whether the linearized MPC still holds the beam; losing stability there would confirm the small-angle assumption as the effective limit.","supporting_citations":[{"cited_title":"Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control,","cited_arxiv_id":null,"evidence_quote":"Supplies the convex model-predictive control formulation for dynamic legged locomotion that this paper adapts to include thruster inputs."},{"cited_title":"Coumans and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the physics simulator in which the beam-walking and push-recovery experiments are run."},{"cited_title":"com / quadprog / quadprog, Version 0.1.11, 2021","cited_arxiv_id":null,"evidence_quote":"Provides the QP solver used to solve the MPC optimization at 100 Hz in the simulations."},{"cited_title":"Generative Design of NU’s Husky Carbon, A Morpho-Functional, Legged Robot,","cited_arxiv_id":null,"evidence_quote":"Establishes the original thruster-assisted legged design and control direction that this work extends to narrow-path walking."},{"cited_title":"Caron, A","cited_arxiv_id":null,"evidence_quote":"Documents the solver benchmark that justifies choosing the QP solver for real-time performance."}],"review_version":1}