Synergistic morphology and feedback control for traversal of unknown compliant obstacles with aerial robots

Dense vegetation is typically perceived by aerial robots as obstacles to avoid, yet branches, twigs, and leaves are compliant and could be traversed by direct physical interaction to access otherwise unreachable areas for environmental monitoring, precision agriculture, and search and rescue. Existing aerial physical interaction (APhI) approaches—such as impedance/admittance control—are primarily designed for exerting forces on rigid surfaces and struggle with the multiple interaction modes required to traverse compliant vegetation (pushing, sliding, and combinations thereof). Model-based or robust controllers can handle complex tasks on rigid or movable structures but rely on high-level, empirically tuned switching conditions that are difficult to define for stochastic, hard-to-model elastic environments like vegetation. Moreover, current drones typically interact via sensorized end-effectors, whereas vegetation traversal requires unconstrained whole-body interactions with distributed haptic sensing. Inspired by animals that synergistically couple morphology and sensory feedback to alternate between interaction modes, this study aims to enable an underactuated quadrotor to traverse a single compliant obstacle with unknown stiffness. The hypothesis is that a task-oriented, streamlined, low-friction body morphology combined with a minimalist feedback controller using force feedback can safely achieve traversal across a wide range of elastic responses without requiring contact models or switching logic.

The paper reviews work on aerial physical interaction where impedance and admittance controls are effective on rigid surfaces but not tailored to compliant, elastic environments (refs. 6–10). Model-based and robust controllers have demonstrated contact, pushing, and sliding in tasks like opening doors and inspection (refs. 11–16), yet these rely on high-level switching policies and well-defined contact models. Extensive APhI research exists for rigid and movable obstacles (ref. 17), whereas elastic obstacles akin to vegetation have received limited attention. A limitation in prior aerial systems is end-effector-centric interaction (refs. 18–23), incompatible with unconstrained whole-body contact needed for vegetation traversal. Biological literature shows animals use streamlined body shapes and sensory feedback to push and slide through clutter (refs. 24–31), inspiring robots with task-oriented morphologies and distributed sensing, including ground robots traversing compliant clutter (refs. 26,27,32), collision-resilient and shape-exploiting drones (refs. 33–36). Prior NMPC and model predictive impedance concepts have been explored for manipulators and aerial robots (refs. 37,38), but not for compliant environments with unknown stiffness. This work addresses these gaps by integrating streamlined morphology, distributed haptics, and an NMPC with impedance behavior to handle unknown elastic interactions without explicit contact models or switching.

Task and environment: The traversal task involves a quadrotor interacting with a single compliant obstacle modeled as a rigid plate hinged vertically with a torsional spring of unknown stiffness K. Interaction is predominantly planar. The drone must follow a straight reference path, make contact, and alternate between interaction modes—Pushing, Sliding, and Push-and-Slide—until detachment and passage past the obstacle.

Robot morphology and sensing: A quadrotor is equipped with a streamlined, disc-shaped (discoid) protective shell with a low-friction fiberglass outer surface. The circular symmetry removes contact discontinuities and facilitates sliding in any in-plane direction. A six-axis force/torque (F/T) load cell between the quadrotor frame and shell measures the net external wrench, effectively providing distributed whole-body haptic sensing (contact can occur anywhere on the shell). Shell construction: laser-cut MDF frame with 3D-printed connectors; fiberglass skin (0.2 mm). A high-friction variant adds Dycem strips; a non-streamlined square cage variant was also manufactured for ablation.

Controller: An NMPC formulated at the drone’s center of mass predicts full rigid-body dynamics including measured external wrench in translational and rotational dynamics. Two main objectives are included: (1) path following along a straight line with desired longitudinal velocity while constraining lateral and vertical drift; (2) a desired impedance behavior during interaction to shape the apparent dynamics and damp oscillations of the elastic environment without an explicit environment model or contact point constraints. The controller balances these objectives via cost weights and runs in a standard feedback loop, sending mass-normalized thrust and attitude references to a low-level attitude controller on the flight controller. Safety constraints on pitch/roll angles and thrust limit exchanged wrenches and prevent oscillations upon contact/detachment; low reference speed ensures quasi-static interaction.

Dynamics and OCP details: State x = [p, v, q, ω], input u = [c, τ]. Discrete-time dynamics are obtained via RK4 from rigid-body equations with external force F_ext and torque τ_ext. The NMPC solves a discretized nonlinear optimal control problem with multi-shooting SQP (ACADO toolkit), including running and terminal costs: lateral/vertical position tracking, velocity regulation (progress along path and zero lateral/vertical velocities), mismatch to desired impedance velocity, yaw regulation, and regularization on angular rates/thrust/torques. Terminal costs penalize lateral/vertical position error and longitudinal velocity error. Constraints include bounds on mass-normalized thrust and on roll/pitch angles to enforce safe interaction. Desired impedance behavior sets apparent mass M ≈ diag(1.2 kg), damping D ≈ I, and zero stiffness K=0, with external force considered on y–z axes to realize predominantly damping behavior, pushing opposite to contact force and indirectly guiding traversal.

Experimental protocol: Hinged plate experiments at three stiffness values spanning one order of magnitude: K = 18, 77.8, and 155.5 N·mm·rad−1 (termed low, mid, high). For each stiffness, 10 trials were conducted at reference speed 0.15 m/s. Metrics computed over the interaction interval (first contact to detachment): mean absolute lateral position error e_p; mean absolute longitudinal velocity error e_v; maximum longitudinal interaction force max(F_ext,x); RMS roll and pitch oscillations. Mann–Whitney U tests compare distributions across stiffness levels. Ablation studies evaluate: (i) a non-streamlined square cage; (ii) a high-friction shell; (iii) controller without haptic wrench feedback and without impedance term (treats interaction as disturbance). Additional experiments traverse real branches (one bare, one with twigs/leaves) anchored at a single point, with the same metrics recorded.

Data/code: Datasets, analysis scripts, and CADs are available at ETH Research Collection (doi:10.3929/ethz-b-000662212); controller code and simulation workspace on Zenodo (doi:10.5281/zenodo.10798275).

• Traversal across unknown stiffness: The drone successfully traversed hinged compliant plates over K = 18, 77.8, 155.5 N·mm·rad−1 with high repeatability. Success rates over 10 trials: low and mid stiffness 10/10; high stiffness 9/10 (one failure).
• Path tracking and speed regulation during interaction: Lateral position MAE remained below ~0.2 m; median e_p across stiffness: 0.084 m (low), 0.098 m (mid), 0.090 m (high). Longitudinal velocity MAE medians: 0.108 m/s (low), 0.104 m/s (mid), 0.109 m/s (high). Mann–Whitney U tests showed no significant differences in e_p or e_v across stiffness (p > 0.05).
• Interaction forces: Max longitudinal force increased with stiffness; medians: 1.193 N (low), 1.272 N (mid), 1.670 N (high). Low vs high distributions differed significantly (Mann–Whitney U, p < 0.05).
• Stability (attitude oscillations): RMS roll medians: 2.664°, 2.100°, 3.525°; RMS pitch medians: 3.935°, 5.057°, 5.828° (all within ±20° bounds). Roll oscillation distributions showed no significant differences (p > 0.05); pitch oscillations differed for low vs mid (p < 0.05) and low vs high (p < 0.01), consistent with greater pitching to generate force at higher stiffness.
• Mode transitions without explicit switching: Observations confirm smooth transitions between Pushing, Sliding, and Push-and-Slide enabled by the streamlined low-friction shell and impedance-modulated NMPC, without contact models or high-level logic. Example mid-stiffness trial shows initial push, then sliding (changes in F_ext,y, yaw ψ, and T_ext,z), followed by push-and-slide with increasing F_ext,x and T_ext,z, and smooth detachment.
• Ablation studies: (i) Non-streamlined square cage: drone remained stuck in Pushing (yaw constrained with the surface), 0% success across stiffness. (ii) High-friction shell: attempted sliding led to aggressive maneuvers and failure to transition to Sliding; 0% success. (iii) Haptic sensing OFF and no impedance term: succeeded at low stiffness via morphology and tracking alone, but failed in 70% (mid) and 100% (high) of trials due to higher wrenches and oscillations, evidencing the necessity of haptic-feedback control for stiffer obstacles.
• Real branches: Successful traversal of single branches (bare and with foliage). Reported metrics (example table across four tests with foliage): e_p = 0.033–0.066 m; e_v = 0.044–0.104 m/s; max F_ext,x = 0.182–0.250 N; RMS yaw = 0.742–1.615°; RMS pitch = 1.134–3.471°. Interactions were shorter and weaker than hinged-plate cases, with quick push-and-slide due to branch snapping away.

The results validate the hypothesis that synergistic integration of a task-oriented, streamlined low-friction body morphology with a minimalist NMPC leveraging haptic-force feedback enables robust traversal of compliant obstacles with unknown elastic responses. The discoid shell allows unconstrained whole-body contact and facilitates sliding, eliminating the need to model contact location or friction cones, while the impedance term within NMPC effectively damps environment-induced oscillations and ensures smooth mode transitions without high-level switching. Performance metrics remain consistent across an order-of-magnitude range of stiffness, with stable attitudes and controlled forces, and generalize to real branches. Ablation studies confirm that morphology alone or control without haptic feedback is insufficient for reliable traversal at higher stiffness: both components are indispensable. These findings indicate a viable path toward embodied APhI where morphology and feedback co-design reduce controller complexity and computational load while enhancing safety and versatility. Potential applications include enabling drones to access cluttered vegetation for ecological monitoring, precision agriculture, and search and rescue, where conventional non-contact flight is limiting.

This work introduces an embodied aerial physical interaction strategy that combines a streamlined, low-friction discoid shell and distributed whole-body haptics with an NMPC incorporating desired impedance behavior to traverse compliant obstacles of unknown stiffness. The approach achieves safe, repeatable traversal across stiffness values spanning an order of magnitude and handles push, slide, and push-and-slide interactions without explicit contact models or switching logic. Experiments with real branches further demonstrate versatility. Future research should: (1) enhance protection and energy absorption with spherical or hemispherical cages to tolerate collisions and relax quasi-static assumptions for higher-speed traversal; (2) increase haptic sensing resolution (e-skins, visual haptics, whiskers) to localize contacts, estimate local compliance, and inform high-level planners to select more traversable paths among multiple obstacles; (3) extend the controller from 2D to full 3D interactions and manage multiple simultaneous contacts; and (4) integrate planning with haptic perception for navigation through dense, heterogeneous vegetation.

The current prototype’s discoid shell leaves propellers exposed to contacts from above and below, limiting robustness in clutter and necessitating quasi-static, low-speed operation. The haptic sensing measures only the net wrench at the body, lacking contact localization and local force estimation, which hinders traversal when multiple obstacles with different stiffnesses are encountered simultaneously; the drone may become stuck on rigid elements without the ability to detect and seek more compliant paths. The controller is formulated for predominantly planar interactions and assumes low-speed, quasi-static conditions; higher speeds would increase wrenches and complicate smooth mode transitions. Contact dynamics are not explicitly modeled; while beneficial for simplicity, this may limit performance in more complex, frictional, or highly nonlinear interactions.