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REVIEW 5 major objections 6 minor 1 cited by

Enabling steep slope walking on Husky using reduced order modeling and quadratic programming

T0 review · 5 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A VLIP model with center-of-mass thrusters plus QP-MPC tracks a 40-degree slope in simulation, and the paper argues the forces transfer to full dynamics.

desk verdict Modest simulation-only VLIP/MPC extension for thruster-assisted steep-slope walking, undermined by an unsupported 'seamless transfer' claim that the paper's own conclusion contradicts. read the letter →

arxiv 2411.11788 v1 pith:SGSKIQUX submitted 2024-11-18 cs.RO cs.SYeess.SY

Kaushik Venkatesh Krishnamurthy , Eric Sihite , Chenghao Wang , Shreyansh Pitroda , Adarsh Salagame , Alireza Ramezani , Morteza Gharib This is my paper · ORCID
classification cs.ROcs.SYeess.SY
keywords variablelengthinvertedpendulumwing-assistedinclinedrunningquadraticprogrammingMPCthruster-assistedlocomotionlegged-aerialrobotfrictionconeconstraintssteepslopewalkingreduced-ordermodel
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 a thruster-assisted legged robot can walk up a 40-degree slope if the control problem is reduced to a planar variable-length inverted pendulum with thruster forces applied at the center of mass and a known, fixed center of pressure. A quadratic-programming model predictive controller solves for ground reaction forces and thruster forces that track a reference trajectory while respecting friction-cone constraints, and the computed forces are said to be transferable to the full Husky dynamics. If correct, this gives a tractable reduced-order control route to wing-assisted-incline-running-style maneuvers on a real legged-aerial platform.

What carries the argument

The central object is a modified Variable Length Inverted Pendulum (VLIP): a point mass connected to a fixed center-of-pressure point on the slope, with an external thruster force at the mass and ground reactions at the center of pressure. The zero-moment-point equation at the center of mass reduces the planar dynamics to the linear time-invariant model in Eq. (13), whose state is $(x_{\text{com}}, \dot{x}_{\text{com}})$ and whose inputs are ground reaction forces. The quadratic-programming MPC with friction-cone constraints is the mechanism that turns this model into a controller, and the paper argues that because the center of pressure can be placed anywhere inside the support polygon by distributing forces between the two contact legs, the thruster wrench from the VLIP can be handed to the full dynamics.

What would settle it

Run the same QP-computed ground reaction and thruster force profiles on the full Husky reduced-order model (HROM) with no-slip feet and posture-based thrust vectoring, and check whether the body follows the reference trajectory; if the full dynamics diverge, or if the required COP lies outside the support polygon, the central transferability claim fails. Alternatively, on hardware, measure whether the actual thruster wrench from the ducted fans matches the commanded $F_x$ and $F_y$ within the friction cone while attempting a 40-degree slope walk.

Watch

Extended reading notes

Core claim

The slope angle drops out of the reduced-order dynamics, leaving a linear time-invariant system whose behavior depends only on the relative position of the center of mass and the center of pressure. Solving the zero-moment-point equation about the center of mass under a constant height $¥ddot{y}=0$ yields $\ddot{x}_{\text{com}} = -(x_{\text{cop}}-x_{\text{com}})\lambda_y/(m y_0) - \lambda_x/m$, and the ground reaction forces $\lambda_x,\lambda_y$ become the inputs. The MPC then chooses these inputs over a horizon to minimize tracking error subject to bounds and the friction cone $\lambda_{y,k} > \lambda_{\min,n}$, $|\lambda_{x,k}| \le \mu_s \lambda_{y,k}$. In simulation on a 40-degree slope the optimizer finds feasible forces, the body tracks the reference position and velocity, and the implied thruster forces point in directions consistent with the wing-assisted inclined running intuition that legs supply traction while thrusters offload normal force.

Load-bearing premise

The center of pressure is assumed known, fixed, and controllable anywhere inside the support polygon, and the commanded thruster forces are assumed physically realizable by pitching the robot body; if either fails on the real Husky, the simulated 40-degree tracking would not transfer to hardware.

Editorial extensions

If this is right

  • The reduced-order model is linear and the QP solver takes about 65 microseconds per call, so the controller is fast enough for real-time implementation on the robot's onboard computer.
  • Ground reaction forces in the VLIP are independent of slope angle, so a single tuning of the MPC could apply to different inclines without re-deriving the model.
  • The framework gives an explicit split of roles: legs provide traction through friction-cone-respecting ground forces, while thrusters offload normal force and help drive acceleration up steep slopes.
  • The same QP formulation can be extended to 3D movements and whole-body control, as the paper identifies as future work.
  • The approach offers a path to dynamic two-point contact gaits on steep slopes, going beyond the static three-foot-contact gaits used by earlier slope-walking controllers.

Reading between the lines

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

  • The fixed-COP assumption is the main gap: the paper does not simulate the full HROM or hardware, so the claim that the thruster wrench 'can be seamlessly transferred' is an extrapolation; I would test robustness by perturbing the COP within the support polygon and measuring tracking degradation.
  • The constant normal force result (normal ground reaction stays at $\lambda_{\min,n}$ because of the quadratic cost) suggests the optimizer actively offloads weight to the thrusters, which implies the achievable range of slope angles is bounded by propeller force limits.
  • A natural extension is to replace the fixed-foothold heuristic with online foothold or step-timing optimization, which would let the same VLIP-QP framework handle irregular terrain and discrete contact switches.
  • The decoupling of slope angle from the reduced-order dynamics means the controller could be reused across terrains without retuning, provided the COP can still be regulated.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper proposes a planar modified Variable Length Inverted Pendulum (VLIP) model with thruster forces applied at the center of mass, and a QP-based MPC that computes optimal ground reaction forces and thruster forces to track a reference trajectory on a 40-degree slope. The authors simulate this reduced-order model in MATLAB, report tracking performance, friction-cone satisfaction, and qpSWIFT solve times, and claim that the resulting forces can be seamlessly transferred to the full Husky dynamics. The paper concludes by listing future work needed for hardware implementation, including kinematic and dynamic constraints, body pitch/roll thrust vectoring, and thrust regeneration via posture manipulation.

Significance. If the reduced-order control framework were validated against the full-order Husky model or hardware, it would be a useful step toward thruster-assisted steep-slope locomotion, building on established VLIP and QP-MPC techniques. The paper's strengths are its simple, real-time-compatible formulation, explicit friction-cone constraints in the optimization, and a slope-independent state model that depends only on the COP-COM offset. However, the current evidence is limited to a simulation of the reduced-order model, in which the thruster forces are computed from the same equations of motion used to propagate the state, so the results demonstrate internal consistency rather than independent predictive validity. The manuscript's own conclusion contradicts the strong transferability claim in Section IV, and the simulation parameters are not fully reported, which limits reproducibility.

major comments (5)
  1. [Section III, Eq. (13)] The linearized dynamics in Eq. (13) are inconsistent with Eq. (12). Using x_cop - x_com = (x_cop,0 - x_com,0) - (x_com - x_com,0), Eq. (12) linearizes to A(2,1) = +λ_y0/(m y0) and B(2,2) = -(x_cop,0 - x_com,0)/(m y0), whereas the paper prints A(2,1) = -λ_y0/(m y0) and B(2,2) = -(x_cop,0 - x_com,0)/y0. The sign error changes the open-loop pole locations, and the missing factor 1/m rescales the control input. Unless a different normalization is intended but not stated, the QP in Section IV is built on a different model than the one derived in Section III.
  2. [Section IV, Eq. (18)] The friction-cone constraint is one-sided. For positive normal forces λ_y,k ≥ λ_minn > 0, the correct no-slip condition is |λ_x,k| ≤ μ_s λ_y,k, i.e., -μ_s λ_y,k ≤ λ_x,k ≤ μ_s λ_y,k. The printed condition λ_x,k < μ_s |λ_y,k| imposes only an upper bound on λ_x and permits arbitrarily large negative tangential forces, which violates the no-slip assumption and undermines the claim that the QP output is constraint-admissible.
  3. [Section IV, last paragraph; Section VI] The claim that the QP-derived thruster forces 'can be seamlessly transferred to the full dynamics' is not supported by the manuscript. The simulation in Section V is run on the planar VLIP model, not on the HROM described in Section II, and no mapping from the commanded ground reaction and thruster forces to joint torques, leg lengths, body orientation, or the thrust-vectoring mechanism is provided. The paper's own conclusion states that 'strictly implementing the kinematic and dynamics constraints would be required,' including body pitch and roll for thrust vectoring and a method of thrust regeneration. The transfer claim should be removed or substantiated with at least a full-HROM simulation.
  4. [Sections III and V] The assumption that the COP is known, fixed, and controllable anywhere within the support polygon is load-bearing but never validated against the four-legged HROM. The MPC commands a scalar offset (x_cop - x_com) through the B matrix, and the resulting thruster forces are two-dimensional; realizing both on the actual robot requires posture manipulation and thrust vectoring that the paper explicitly defers to future work. The simulation therefore demonstrates internal feasibility of the reduced-order model, not feasibility on Husky.
  5. [Section V] The simulation results are not reproducible as reported. The MPC weights Q and R, prediction horizon n_h, discretization time step, λ_minn, μ_s, the reference trajectory, and the COP update rule are not specified; only the slope angle and the solver's average solve time are given. Without these parameters, the tracking and constraint-satisfaction results in Figs. 4-7 cannot be independently checked.
minor comments (6)
  1. [Section I] The paper structure says 'In Section 1... Section 2...' but the actual sections are labeled with Roman numerals II-V; renumber the outline accordingly.
  2. [Section IV] The constraint line 'Umin and Umin' should read 'Umin and Umax'.
  3. [Fig. 6 caption] The caption says the forces are 'from (9)', but the thruster forces are computed from Eq. (10); correct the equation reference.
  4. [Fig. 7] The figure plots λ1 and λ2 but does not define these variables in the text or caption; define them.
  5. [Throughout] The spacing in 'W AIR' is inconsistent; use 'WAIR' consistently, and fix the incomplete sentence ending 'friction cone constraints and.' in Section I.
  6. [Fig. 4] The title contains a typo: 'referance' should be 'reference'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the VLIP/QP simulation is self-contained, and the unsupported full-dynamics transfer claim is an extrapolation rather than a definitional reduction.

full rationale

The derivation chain is not circular. The modified VLIP model in Section III is derived from Newtonian mechanics with explicitly stated assumptions: constant COM height y0, known and controllable COP, and thruster forces collocated at the COM. The QP MPC in Section IV minimizes a tracking cost subject to input bounds and friction-cone constraints, and the simulation in Section V is an internal feasibility check on that model. No parameter is fitted to the target result; the reference trajectory and COP update rule are hand-chosen, which limits external validity but does not make the output equivalent to the input by construction. The thruster forces are indeed computed from the same equations of motion (Eq. 10) after the QP selects ground reactions, but that is standard inverse-dynamics consistency for a feasibility study, not a fitted prediction. The statement that these forces 'can be seamlessly transferred to the full dynamics' (Section IV) is unsupported and is contradicted by the paper's own conclusion, which defers kinematic and dynamic constraints, pitch and roll enforcement, and thrust-regeneration mechanisms to future work; that is a validation gap or overclaim, not circularity. Self-citations to Husky Carbon prior work establish hardware context and are not load-bearing for the model or MPC derivation. There is no imported uniqueness theorem and no ansatz smuggled in via citation. Therefore the circularity score is 0.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The model relies on a set of hand-chosen parameters and idealizations: a fixed and controllable COP, constant COM height, massless legs, and a thruster wrench at the COM. The linearization points (lambda_y0, xcop,0 - xcom,0), MPC weights, and the reference trajectory/COP heuristic are not disclosed with numbers, so the simulation result is conditioned on unstated tuning choices.

free parameters (7)
  • minimum normal force lambda_minn = not specified
    Constraint lower bound for normal ground reaction force, chosen by hand in Section IV; it directly affects the normal force and thruster force magnitudes.
  • friction coefficient mu_s = 0.5
    Set in Section V for the friction cone constraint; no justification or sensitivity analysis.
  • COM height y0 = not specified
    Assumed constant in the LIP model (Section III); an input parameter to the linearized dynamics.
  • nominal normal force lambda_y0 = not specified
    Used in the linearization A matrix in (13); chosen without explanation.
  • nominal COP-COM offset (xcop,0 - xcom,0) = not specified
    Used in the B matrix in (13); a linearization point chosen without discussion.
  • MPC weights Q and R = not specified
    Cost function weights in the QP (Section IV) are not given, yet they determine the optimal force distribution.
  • reference trajectory and COP update rule = hand-designed
    Selected as a heuristic so body motion is symmetric about the COP (Section V); this is effectively tuned to make the simulation succeed.
assumptions (6)
  • domain assumption Massless legs with no-slip contact
    Introduced in Section II, the HROM treats legs as massless and enforces no-slip; this removes leg dynamics from the model.
  • domain assumption Planar motion confined to the sagittal plane
    Section V constrains the frontal plane and evaluates only the planar model, ignoring roll and lateral dynamics.
  • domain assumption Constant COM height, yddot = 0
    Section III assumes the perpendicular distance from the slope is constant, which is standard for LIP but may not hold during dynamic gait.
  • ad hoc to paper COP is known and controllable within the support polygon
    Section III states this assumption without deriving how two-leg force distribution achieves it; the full dynamics are not simulated to verify.
  • domain assumption Thruster wrench collocated at the COM
    Section II defines u_e as the external thrust wrench acting on the COM; this ignores the physical mounting and thrust-vectoring kinematics.
  • domain assumption Friction cone constraints are sufficient for no-slip
    Section IV uses a friction cone condition; the one-sided inequality in (18) is itself questionable.

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Cite this review

Pith. "Pith review of Enabling steep slope walking on Husky using reduced order modeling and quadratic programming." pith.science (2026). https://pith.science/paper/SGSKIQUX

@misc{pith2026241111788,
  author       = {Pith},
  title        = {Pith review of: Enabling steep slope walking on Husky using reduced order modeling and quadratic programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGSKIQUX}},
  note         = {Machine review of arXiv:2411.11788}
}
read the original abstract

Wing-assisted inclined running (WAIR) observed in some young birds, is an attractive maneuver that can be extended to legged aerial systems. This study proposes a control method using a modified Variable Length Inverted Pendulum (VLIP) by assuming a fixed zero moment point and thruster forces collocated at the center of mass of the pendulum. A QP MPC is used to find the optimal ground reaction forces and thruster forces to track a reference position and velocity trajectory. Simulation results of this VLIP model on a slope of 40 degrees is maintained and shows thruster forces that can be obtained through posture manipulation. The simulation also provides insight to how the combined efforts of the thrusters and the tractive forces from the legs make WAIR possible in thruster-assisted legged systems.

Figures

Figures reproduced from arXiv: 2411.11788 by the authors.

Figure 1
Figure 1. Image of Husky Carbon [4], a legged-aerial robot, positioned on [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Depicts the parameters of the full, Husky reduced-order model (HROM), and the planar variable length inverted pendulum (VLIP) model, which [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure depicting Snapshots from the Matlab simulation of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Error and tracking performance of body position and velocity [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Position tracking in COP frame Thruster Forces & 6 Time (s) 0 1 2 3 4 5 6 7 8 9 10 -50 0 50 100 N Fx Fy 0 1 2 3 4 5 6 7 8 9 10 -50 0 50 N 6x 6y [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Calculated optimal ground reaction forces and thruster forces from [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: Friction cone constraint satisfaction of the ground reaction forces [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: qpSWIFT solve times for a MPC horizon of 5 and step time of [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

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Forward citations

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.