Shape-Adaptive Planning and Control for a Deformable Quadrotor

Fei Gao; Junjie Wang; Xuankang Wu; Yuan Zhou; Yuze Wu; Zheng Fang; Zhichao Han

REVIEW 4 major objections 5 minor 23 references

Shape-Adaptive Planning and Control for a Deformable Quadrotor

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A deformable quadrotor can plan its own morphing schedule and track it accurately enough to slip through gaps smaller than its full frame.

desk verdict Solid systems paper on shape-adaptive planning for a morphing quadrotor; the planning idea is genuinely new and the real experiments are real evidence, but the headline tracking claim and the energy comparison rest on thinner support than the text suggests. read the letter →

arxiv 2505.15010 v1 pith:JB7NUKA2 submitted 2025-05-21 cs.RO

Yuze Wu , Zhichao Han , Xuankang Wu , Yuan Zhou , Junjie Wang , Zheng Fang , Fei Gao This is my paper · ORCID

classification cs.RO

keywords deformablequadrotorshape-adaptiveplanningkinodynamicA*trajectoryoptimizationexternalforcecompensationincrementalnonlineardynamicinversionnarrow-gaptraversalwhole-bodygrasping

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 tries to establish that a quadrotor whose arms can retract can be given full autonomy: a planner that treats the deformation radius as a state co-designs the flight path and the morphing schedule, and a controller that estimates and cancels external forces and torques lets the drone track those shape-changing trajectories precisely. If true, a deformable drone can choose its size deliberately, staying large for energy efficiency and control authority and shrinking only when a gap demands it. The authors validate this in simulation and on hardware, reporting a 37.3 percent reduction in trajectory tracking error over their earlier controller.

What carries the argument

The paper's load-bearing object is the trajectory optimization of Eq. (13), which minimizes the sum of position control effort, radius control effort, a second-order radius regularization term that penalizes deviation from maximum size, and flight time, subject to velocity and acceleration bounds on both position and radius and a full-shape safety constraint. The front-end scalable kinodynamic A* search uses a state space augmented with radius and its derivative, so the planner can decide when to shrink. The servo is modeled as a first-order system with a single time constant, and the controller compensates for disturbances through Eqs. (23) through (25).

What would settle it

Fly the drone through an aggressive planned morphing maneuver while independently measuring the actual radius; if the radius error exceeds the assumed bounds or the thrust coefficient changes by more than the model allows, the decoupled model is falsified.

Watch

Extended reading notes

Core claim

The central claim is that a single optimization over position and radius, subject to full-shape collision constraints and dynamic feasibility, produces trajectories that outperform both fixed-maximum-size and fixed-minimum-size planning in total cost. The authors further claim that their controller, combining nonlinear model predictive control with external-force compensation in the thrust loop and incremental nonlinear dynamic inversion torque compensation in the attitude loop, achieves the reported 37.3 percent tracking improvement. This is a new application of planning and control to a deformable platform, and the key is treating the deformation radius as a controllable state rather than a fixed geometry.

Load-bearing premise

The pipeline treats the deformation radius as an independently controllable double-integrator state that can track any commanded profile within rate limits, without explicitly modeling the coupling between arm retraction and rotor aerodynamics.

Editorial extensions

If this is right

The planner extends naturally to carrying grasped objects by adding the object's volume to the collision model, enabling whole-body transport through irregular openings.
If the radius model holds, the same front-end and back-end structure could be applied to other shape-changing robots, such as foldable or tilting-rotor platforms.
The force and torque compensation scheme may generalize to any quadrotor experiencing changing inertia or external loads, not only deformable designs.
The reported 37.3 percent error reduction is measured against the authors' prior controller; the approach should also reduce error under wind or payload changes, though that is not explicitly tested here.

Reading between the lines

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

A natural extension is to learn a radius-dependent thrust map that captures blade-overlap and airflow effects, and to feed that map into both the planner's cost and the controller's model.
The double-integrator model for radius is a deliberate simplification; the framework would likely survive a more accurate high-order servo model as long as the planner's constraints are updated.
A testable extension is running the planner in an environment with moving narrow gaps to see whether the radius regularization term helps the search escape local minima and maintain passability.
The reported tracking improvement is for a figure-eight trajectory at up to 1.5 m/s; it remains open whether the compensation holds at more aggressive speeds, where aerodynamic coupling is stronger.

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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

Short version: worth reading and worth a referee. The new contribution is the shape-aware front-end: treating the deformation radius r as an explicit state in a kinodynamic A* search, then co-optimizing position and radius in the MINCO-style backend with full-shape safety constraints. That goes beyond Cui et al., which plans for morphing drones but relies on manual or point-mass graph searches. The extension to include a grasped object in the collision model is clean and useful. The control part—NMPC plus external-force compensation and INDI torque compensation—is standard machinery but sensibly integrated, and the real flights through a 40 cm gap and an 8 cm-overwidth cross gap with a 20 cm object are genuine evidence that the system works. The math is standard and the paper is clearly written.

The soft spots are real but not fatal. First, the headline 37.3% tracking-error reduction comes from one figure-eight trial with no error bars. The prior [6] RMSE and the new RMSE are single numbers. That is not enough to support a quantitative headline. Second, the energy comparison in Table I is partly circular: the cost in Eq. (13) includes a(r-rmax)^2, and the 'total energy' model from [6,23] also integrates (r-rmax)^2. So the planner is being rewarded for exactly the quantity the energy metric counts; the comparison still shows sensible behavior, but it is not an independent check. Third, the planning model treats r as a double-integrator state with rate and acceleration limits, while the hardware is a single tendon-driven servo approximated as first-order with time constant gamma. The paper never shows the commanded radius versus the measured radius during flight, and never states the safety margin D_margin. Since the 40 cm gap is only 2.2 cm narrower than the maximum size, an unverified radius lag could matter. I do not think this sinks the contribution—the experiments did succeed—but it does mean the safety and transferability claims are under-supported.

The citation pattern is fine; the heavy reliance on [6] is legitimate since it is the same platform, and the comparison to Cui et al. is accurately drawn. Overall, this is a competent systems paper that deserves serious peer review. I would ask for more trials with variance, the actual parameter values, radius tracking data, and preferably code or data release before accepting. It is not desk-reject material.

Referee Report

4 major / 5 minor

Summary. The paper presents a planning-and-control framework for a deformable quadrotor. The planner augments kinodynamic A* with a deformation-radius state and then solves a spatio-temporal trajectory optimization over centroid position and radius, subject to full-shape collision constraints. The controller adds external-force compensation (Section IV-C.1) and INDI torque compensation (Section IV-C.2) on top of an NMPC baseline. The paper claims a 37.3% reduction in tracking RMSE relative to the authors' prior work, and demonstrates real-world narrow-gap traversal and whole-body grasping-transport through a cross-shaped gap. Validation includes a simulation cost comparison (Table I) and several hardware experiments.

Significance. If substantiated, the paper offers a practically useful integration of shape-change planning and disturbance-compensated control for morphing multirotors. The strengths are the physical hardware demonstrations, the explicit co-optimization formulation of Eq. (13), and the real-world traversal of a 40-cm gap and an 8-cm-overwidth cross-shaped gap. However, the headline tracking-improvement number rests on a single trial, the energy advantage is partly built into the optimization objective, and the decoupled radius model used in planning is not validated against measured morphing dynamics. These gaps mean the central claims are plausible but not yet fully supported.

major comments (4)

[V-C.3, Fig. 8] The 37.3% RMSE improvement (0.0561 m to 0.0352 m) is reported from a single figure-eight experiment with no error bars, trial count, or statistical test. Because this is the paper's headline quantitative claim, please repeat the experiment multiple times and report mean +/- standard deviation, the number of trials, and the experimental conditions. Without this, the improvement cannot be distinguished from run-to-run variability.
[IV-B Eq. (13), V-B, Table I] The 'Total Energy' metric used to support the energy-efficiency advantage is computed with a power model that includes a (r(t)-rmax)^2 term, the same term that appears in the planning objective (Eq. 13). Hence part of the energy benefit of the adaptive planner is enforced by construction rather than discovered. Please evaluate energy with an independent model (for example, from measured rotor speeds or electrical power) or explicitly state that Table I compares design-objective costs rather than physical energy consumption.
[III, IV-A, IV-D, Eq. (4), (16), (27)] The planner models the deformation radius as an independent double-integrator state with hard velocity and acceleration limits, while the real platform uses one tendon-driven servo with first-order lag (Eq. 27), and morphing changes motor-arm geometry, inertia, and airflow. No experiment measures the actual radius trajectory during gap traversal, so it is unverified that the planned r(t) is trackable within the safety margins of Eq. (17). Since the 40 cm gap is only 2.2 cm smaller than the maximum size, a small lag or overshoot in the radius could violate the clearance. Please provide measured radius or actuator-angle data during a traversal, or a margin analysis that accounts for the first-order servo dynamics.
[V-B, Table I] The cost comparison claims 'average metric costs from hundreds of planning runs' but reports only point values with no standard deviations, number of runs, or significance test. This weakens the claim of 'superior performance' in total cost. Please report variance or at least state explicitly that the differences are qualitative.

minor comments (5)

[V-C.3 and Fig. 8 caption] Use 'figure-eight' rather than '8-figure' consistently in the text and caption.
[Eq. (24)] Define F_u clearly; since u in Eq. (22) treats F_u as a scalar thrust command, the expression Fdes = ||FuzB - Fext|| should explain how the scalar and vector quantities are combined.
[Eq. (14) and Eq. (18)] The notation sigma[s-1] and sigma[ed] is used before it is defined; please introduce these superscripts explicitly.
[V-B] Provide the explicit power-model equation used for 'Total Energy' in Table I, including the coefficients and the exact combination of ||F||^(3/2) and (r-rmax)^2 terms; the current verbal description is imprecise.
[IV-A and Table I] In Eq. (7), the cost includes a(r-rmax)^2*DeltaT, while the table reports SORR in units consistent with an integral of squared radius acceleration; clarify how the 'second-order radius regularization' in Eq. (13) relates to the SORR column in Table I.

Circularity Check

1 steps flagged · score 4.0 of 10

Energy evaluation reuses the planner's own radius-regularization term; tracking and traversal results remain independent.

self definitional [Section V-B (Simulation Experiments), Table I; objective in Section IV-B, Eq. (13)]
"we calculate the total energy consumption of flight based on the power model [6, 23], which is in the form of the time integral of ||F||^(3/2) and ( (r(t)-r_max)/r_max )^2. ... min_{c,T} J = ∫_0^T [ p_W^{(s)T} p_W^{(s)} + (r^{(s)})^2 + a( (r(t)-r_max)/r_max )^2 ] dt + w_T T"

Eq. (13) explicitly minimizes the second-order radius regularization term a((r(t)-r_max)/r_max)^2 as part of the trajectory objective. The 'Total Energy' column of Table I is then computed from a power model whose radius-dependent part is exactly ((r(t)-r_max)/r_max)^2. Consequently, any trajectory optimized with Eq. (13) is scored lower on that component of the energy metric by construction; in particular, the comparison against fixed-min-size planning, which must hold r = r_min and therefore incur a large radius penalty, is not an independent energy measurement. The thrust term ||F||^{3/2} in the power model is not identical to the position control effort p^{(s)T}p^{(s)}, so the circularity covers only the radius portion of the reported energy advantage.

full rationale

The trajectory derivation is self-contained: Eq. (13) optimizes position and radius control effort, a radius-regularization penalty, and time subject to dynamic-feasibility constraints (Eqs. 15-16) and full-shape safety constraints (Eq. 17), with the front-end kinodynamic A* providing a warm start. The 37.3% tracking-error improvement is an independent experimental comparison against the authors' previous controller [6], executed on the same platform with a figure-eight reference, so it does not reduce to the planning objective. The narrow-gap traversal and grasping-transport demonstrations are direct empirical outcomes that do not depend on the energy metric. The only genuine circularity is in the simulation energy evaluation: the 'Total Energy' metric is defined with the same ((r-rmax)/rmax)^2 radius penalty that Eq. (13) explicitly minimizes, so part of the reported energy advantage over fixed-min-size planning is enforced by the objective rather than discovered. The ||F||^{3/2} portion is not the same as the position control effort, so the overlap is partial. The decoupled double-integrator radius model and first-order servo approximation are validity and robustness concerns, not circularity, and do not raise the score.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests mainly on hand-tuned planner and controller weights and on modeling assumptions about the robot's geometry and morphing dynamics; no new physical entities are proposed.

free parameters (6)

a = not reported
Weight on the second-order radius regularization term in Eq. (7) and Eq. (13); it controls how strongly the planner prefers large radius and directly shapes the energy-related cost.
w_T = not reported
Weight on flight time cost in Eq. (7) and Eq. (13); tunes how aggressively the planner minimizes duration versus smoothness and size.
D_margin = not reported
Safety clearance in Eq. (6) and Eq. (17); hand-set threshold that determines how close the robot may pass to obstacles.
NMPC weights Q, Q_N, W = not reported
Weight matrices for state and input costs in Eq. (22); hand-tuned and not specified, so controller behavior is not reproducible from the paper.
Servo time constant gamma = not reported
Time constant of the first-order servo model in Eq. (27); must be identified from hardware but no value or identification procedure is given.
Dynamic limits v_max, a_max, omega_max, alpha_max = not reported
Feasibility bounds in Eqs. (15) and (16); chosen from hardware intuition and not reported, affecting whether planned trajectories are actually executable.

assumptions (5)

domain assumption The robot's collision geometry is a cylinder with radius r and fixed height h, and deformation is captured by a single scalar radius r.
Invoked in Eq. (20) and (21) for sampled surface points; the real ring-shaped quadrotor has four arms and may not be exactly cylindrical during morphing or grasping.
domain assumption The deformation radius r evolves as a double integrator with bounded velocity and acceleration, decoupled from translational and rotational dynamics.
State transition in Eq. (4) and constraints in Eq. (16) assume r can be commanded freely within limits; hardware uses a single tendon actuator and morphing affects aerodynamics and inertia.
domain assumption The environment is represented by an accurate, static Euclidean signed distance field built from radar and motion capture data.
Safety constraints Eq. (6) and (17) require the ESDF to reflect true obstacle distances; map error or dynamic obstacles are not handled.
domain assumption The power/energy model used for comparison is a time integral of ||F||^(3/2) and (r-rmax)^2, as stated in Section V-B.
This model, adopted from [6,23], assumes energy scales with thrust power and with the radius penalty; if the model is inaccurate, the energy comparison is not a physical measurement.
standard math Kinodynamic A*, MINCO reparameterization, NMPC, and INDI are assumed correct as established methods.
Used as building blocks without re-derivation; standard in the field.

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Cite this review

Pith. "Pith review of Shape-Adaptive Planning and Control for a Deformable Quadrotor." pith.science (2026). https://pith.science/paper/JB7NUKA2

@misc{pith2026250515010,
  author       = {Pith},
  title        = {Pith review of: Shape-Adaptive Planning and Control for a Deformable Quadrotor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JB7NUKA2}},
  note         = {Machine review of arXiv:2505.15010}
}

read the original abstract

Drones have become essential in various applications, but conventional quadrotors face limitations in confined spaces and complex tasks. Deformable drones, which can adapt their shape in real-time, offer a promising solution to overcome these challenges, while also enhancing maneuverability and enabling novel tasks like object grasping. This paper presents a novel approach to autonomous motion planning and control for deformable quadrotors. We introduce a shape-adaptive trajectory planner that incorporates deformation dynamics into path generation, using a scalable kinodynamic A* search to handle deformation parameters in complex environments. The backend spatio-temporal optimization is capable of generating optimally smooth trajectories that incorporate shape deformation. Additionally, we propose an enhanced control strategy that compensates for external forces and torque disturbances, achieving a 37.3\% reduction in trajectory tracking error compared to our previous work. Our approach is validated through simulations and real-world experiments, demonstrating its effectiveness in narrow-gap traversal and multi-modal deformable tasks.

Figures

Figures reproduced from arXiv: 2505.15010 by the authors.

**Figure 1.** (a) Shape-adaptive motion planning for deformable traversal and whole-body grasping-transport. (b) Highlight of autonomous [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** (a-b) Mechanical schematics of the morphing quadrotor named RR: the robot utilizes a single motor and tendon-driven system for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Comparative schematic diagram of front-end kinodynamic [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The schematic diagram of the proposed enhanced adaptive [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: A comparison of trajectories between the proposed method and traditional fixed-size motion planning approaches [ [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: The experiment of autonomously navigating narrow and complex spaces with deformation. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: The autonomous experiment on whole-body grasping-transport with large objects through irregular environments. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: Enhanced control performance in tracking 8-figure trajecto [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 7, 2026 · model on record in the stance chip above.

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