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Temporal Learning for End-Effector Position Estimation under Aerodynamic Disturbances in Aerial Continuum Manipulation

Niloufar Amiri, Houman Masnavi, Farrokh Janabi-Sharifi

arXiv:2609.28716v1cs.ROcs.AIcs.LG

TL;DR

UAV-induced aerodynamic effects can shift the end-effector position of compliant aerial continuum manipulators, while nominal strain-model accuracy trades off against spatial complexity. The paper establishes a strain-parameterized baseline and compares static, recurrent, and continuous-time residual estimators; on unseen experiments, CfC achieves the lowest reported 3D RMSE among them.

Abstract

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This paper investigates temporal neural networks for \mbox{end-effector} position \mbox{estimation} of an aerial continuum manipulator (ACM) operating under aerodynamic effects induced by the unmanned aerial vehicle (UAV). An experimental dataset is collected under stationary (\mbox{rotor-off}) and \mbox{free-hovering} conditions across continuum robot (CR) configurations and UAV altitudes, providing \mbox{end-effector} position measurements with and without aerodynamic residuals. To establish a nominal framework, \mbox{strain-parameterized} kinematic models with progressively richer strain bases are evaluated to balance model complexity and prediction accuracy. The selected nominal model then serves as the baseline for 3D position residual estimation using a \mbox{closed-form} \mbox{continuous-time} (CfC) neural network, with a multilayer perceptron (MLP) and a gated recurrent unit (GRU) used for comparison. On unseen test experiments, the CfC achieves an RMSE of \(22.00\pm1.70~\mathrm{mm}\) over five random seeds, compared with \(36.38\pm3.58~\mathrm{mm}\) for the MLP and \(27.72\pm2.92~\mathrm{mm}\) for the GRU, corresponding to reductions of \(39.52\%\) and \(20.62\%\), respectively. These results demonstrate the effectiveness of \mbox{continuous-time} learning for \mbox{end-effector} position estimation under aerodynamic disturbances relative to static and \mbox{discrete-time} learning methods.

I. INTRODUCTION

Aerial continuum manipulators are susceptible to UAV-induced aerodynamic disturbances that can shift CR configuration and end-effector position. The paper addresses nominal-model complexity and temporal residual estimation, extending prior steady-state correction toward free-hovering conditions.

I. INTRODUCTION

  • UAV-induced aerodynamic disturbances can measurably alter continuum-robot configuration and end-effector position in lightweight, compliant aerial continuum manipulators.Source: §I · p. 1
  • Nominal strain-basis order trades prediction accuracy against added generalized coordinates and model complexity, so higher-order models should justify their complexity with meaningful gains.Source: §I · p. 1
  • Configuration-dependent downwash and motion-history effects motivate temporal models, with continuous-time networks representing hidden-state evolution alongside recurrent memory.Source: §I · p. 1

II. RELATED WORK

  • The study extends steady-state correction by evaluating static, discrete-time recurrent, and continuous-time models for 3D residual estimation with a free-hovering UAV.Source: §II · p. 2
  • The contributions include an experimental dataset, a comparative study of strain-basis orders, and systematic MLP, GRU, and CfC residual-estimation comparisons.Source: §II · p. 2

III. PROBLEM FORMULATION AND METHODOLOGY

The nominal kinematic model parameterizes continuum-robot strain with spatial bases and maps actuator coordinates to end-effector position. Its linear strain basis is selected as a compact representation and calibrated using stationary baseline measurements before residual learning.

A. Nominal Strain-Parameterized Model

  • The CR configuration is represented along arc length by cross-section orientation and position evolving according to strain-parameterized Cosserat rod kinematics.Source: A. Nominal Strain-Parameterized Model · p. 2
  • The distributed strain field is approximated using generalized strain coordinates and a spatial basis whose order determines the parameterization.Source: A. Nominal Strain-Parameterized Model · p. 2
  • Only two bending strains are parameterized, while torsion, extension, and shear remain at reference values; shifted Legendre polynomials provide the spatial basis.Source: A. Nominal Strain-Parameterized Model · p. 2
  • Constant through cubic shifted-Legendre spatial bases are evaluated, and the linear basis is selected to balance nominal prediction accuracy and model complexity.Source: A. Nominal Strain-Parameterized Model · p. 2
  • For the selected linear model, bending amplitudes vary spatially according to first-order shifted Legendre terms, with amplitudes related to actuator coordinates through a separately calibrated mapping.Source: A. Nominal Strain-Parameterized Model · p. 2
  • The model obtains nominal end-effector position from the translational component at the tip and expresses that prediction in the experimental measurement frame.Source: A. Nominal Strain-Parameterized Model · p. 2
  • Spatial parameters are identified by minimizing position error on stationary measurements, and the resulting nominal model is fixed before aerodynamic residual learning.Source: A. Nominal Strain-Parameterized Model · p. 2
  • The illustrated platform depicts ACM frame attachment and operating conditions across different continuum-robot configurations.Source: Figure 1 · p. 3

B. UAV-Induced Residual Estimation

The residual estimator predicts the difference between free-hovering measurements and the stationary nominal response. Its inputs include actuation, altitude, and throttle signals, while the residual is interpreted as an effective position deviation rather than a direct aerodynamic-force estimate.

B. UAV-Induced Residual Estimation

  • The estimator uses actuator coordinates, reciprocating actuation rate, UAV hovering altitude, and recorded throttle to estimate free-hovering deviations from the stationary nominal response.Source: B. UAV-Induced Residual Estimation · p. 3
  • The 3D position residual is defined relative to the nominal-model prediction and is estimated from the neural-network inputs and preceding observations.Source: B. UAV-Induced Residual Estimation · p. 3
  • Because the residual includes possible nominal-model mismatch as well as UAV-induced deviations, it is an effective position residual, not a direct aerodynamic-force measurement.Source: B. UAV-Induced Residual Estimation · p. 3

C. Comparative Neural Architectures

The section compares a memoryless MLP, a discrete-time recurrent GRU, and a CfC with a closed-form state transition derived from continuous-time dynamics. Recurrent inputs use finite sequences, with all models evaluated at identical target timestamps.

C. Comparative Neural Architectures

  • The MLP is memoryless, the GRU uses discrete-time recurrent memory, and the CfC uses a closed-form transition derived from continuous-time neural dynamics.The CfC transition avoids numerical integration of the underlying state evolution.Source: C. Comparative Neural Architectures · p. 3
  • GRU and CfC inputs are arranged as finite sequences of consecutive measurements ending at sample k.Source: C. Comparative Neural Architectures · p. 3
  • Sequences are constructed independently within each experiment, and all three models are evaluated at identical target timestamps.A gap exceeding the prescribed ∆tmax starts a new sequence segment.Source: C. Comparative Neural Architectures · p. 3

1) Multilayer Perceptron:

The MLP estimates the residual from the instantaneous input vector using a static mapping without an explicit recurrent state.

1) Multilayer Perceptron:

2) Gated Recurrent Unit:

The GRU uses a recurrent hidden state to retain information from preceding observations and maps that state to a residual estimate.

2) Gated Recurrent Unit:

3) Closed-Form Continuous-Time Network:

The CfC uses a closed-form recurrent transition derived from continuous-time neural dynamics and accounts explicitly for elapsed time between measurements.

3) Closed-Form Continuous-Time Network:

IV. EXPERIMENTAL SETUP AND DATASET GENERATION

The dataset combines rotor-off baseline experiments with free-hovering trials across continuum-robot configurations and UAV altitudes. Measurements are split by experiment, with separate data reserved for nominal-model calibration and unseen evaluation.

IV. EXPERIMENTAL SETUP AND DATASET GENERATION

  • Twelve q1 workspace slices are covered by three experiments per hovering altitude, with q1 held constant and q2 reciprocating in each experiment.Experiments are repeated at four altitudes for 12 free-hovering experiments; five additional baseline experiments use inactive rotors.Source: §IV · p. 4
  • The CR moves slowly enough to treat its motion as kinetostatic, while a Vicon system records 3D positions at 10 Hz.UAV throttle and the two tendon-actuating servo encoder measurements are also recorded.Source: §IV · p. 4
  • The dataset contains 28,411 samples: 10,538 baseline and 17,873 under UAV operation, with payload tests excluded from residual estimation.Domain filtering retains 13,922 free-hovering samples; Tests 1–9 form development data, Tests 10–12 unseen evaluation, and Tests 17–21 nominal calibration.Source: §IV · p. 4

V. RESULTS AND ANALYSIS

The linear strain basis provides a compact nominal representation, sharply reducing error over the constant-strain model; the selected strain model is comparable in accuracy to a third-order polynomial with fewer parameters. Aerodynamic residuals vary with altitude and configuration, occur in all Cartesian directions, and are largest along z.

B. UAV-Induced Residual Characterization

  • 44.05 mm overall 3D RMS residual across Tests 1–12; residuals vary nonmonotonically with altitude and nonlinearly across configuration slices.Residuals occur in all three Cartesian directions, with the largest RMS component along z, consistent with the primary direction of rotor-induced downwash.Source: B. UAV-Induced Residual Characterization · pp. 4–5

A. Nominal Model Evaluation

  • 13.84 mm binned x–z RMSE for the linear basis versus 94.72 mm for constant strain on Test 17; quadratic and cubic bases reach approximately 13.03 mm.The linear basis is selected as a compact representation because higher-order bases provide only marginal further improvement.Source: A. Nominal Model Evaluation · p. 4, Figure 2 · p. 5
  • 25.19 mm 3D RMSE for the strain model versus 24.87 mm for the third-order polynomial, using 20 versus 30 identified parameters.The strain model preserves a physically structured representation of distributed deformation rather than directly fitting Cartesian tip coordinates.Source: A. Nominal Model Evaluation · p. 4, Figure 2 · p. 5

C. Learning-Based Residual Estimation

On unseen experiments, temporal residual estimators improve on the nominal reference, with CfC leading across five-seed evaluation. Input ablations show that actuation rate and altitude contribute to prediction accuracy under the tested conditions.

C. Learning-Based Residual Estimation

  • Three-fold cross-validation selects sequence lengths of 25 and 50 samples; each fold validates on one experiment from each development altitude and trains on the remaining experiments, representing all development altitudes in both partitions.Standardization statistics are computed exclusively from each training partition, and the unseen dataset is reserved for final evaluation.Source: C. Learning-Based Residual Estimation · p. 5
  • On the unseen dataset, 3D RMSE is 21.87 mm for CfC, 27.75 mm for GRU, 41.98 mm for MLP, and 44.94 mm for the nominal model.The unseen experiments were collected at a hovering altitude excluded from neural-network development.Source: C. Learning-Based Residual Estimation · p. 5, Figure 4 · p. 7
  • CfC residual R2 values are approximately 0.75, 0.39, and 0.67 for ∆x, ∆y, and ∆z; MLP has negative R2 for ∆x and ∆z, while GRU is positive in all directions.The time histories show generally lower instantaneous 3D errors for temporal models, but performance varies by condition and Test 12 is most challenging.Source: C. Learning-Based Residual Estimation · pp. 5–6

D. Model Sensitivity and Robustness

  • Removing actuation rate raises 3D RMSE by 11.34% for CfC and 11.20% for GRU; removing altitude raises it by 18.89% and 7.52%, while removing throttle slightly lowers validation RMSE.The tested throttle range was relatively narrow because the UAV operated near its available thrust capacity.Source: D. Model Sensitivity and Robustness · p. 6, Table I · p. 7
  • On the unseen dataset over five random seeds, CfC achieved 22.00 ± 1.70 mm versus GRU’s 27.72 ± 2.92 mm, reducing mean 3D RMSE by approximately 39.5% versus MLP and 20.6% versus GRU.CfC also showed the smallest variation in 3D RMSE across the five seeds and the lowest 3D error in each run.Source: D. Model Sensitivity and Robustness · p. 7, Table II · p. 8

VI. CONCLUSION

The conclusion identifies modeling complexity and UAV-induced aerodynamic effects as challenges for aerial continuum manipulators. It reports a structured compact strain model and improved unseen-experiment estimation with temporal residual learning, while reserving more aggressive maneuvers and stochastic aerodynamics for future work.

VI. CONCLUSION

  • A compact physics-informed strain model provides accuracy comparable to a data-driven polynomial model while retaining a more structured representation.The conclusion frames reduced-order models as attractive for real-time deployment while emphasizing the accuracy–complexity trade-off.Source: §VI · p. 8
  • Real-world experiments reveal noticeable end-effector position deviations under UAV-induced aerodynamic effects, and temporal models outperform a memoryless approach on unseen experiments.The paper presents the approach as lightweight and deployable for improving ACM end-effector position estimation.Source: §VI · p. 8
  • Future work will extend the framework to more aggressive UAV maneuvers and stochastic aerodynamic conditions.Source: §VI · p. 8
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