Paper deep dive
Temporal Learning for End-Effector Position Estimation under Aerodynamic Disturbances in Aerial Continuum Manipulation
Niloufar Amiri, Houman Masnavi, Farrokh Janabi-Sharifi
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
Last extracted: 9/25/2026, 4:08:48 AM
Summary
This paper investigates the use of temporal neural networks for estimating the end-effector position of an aerial continuum manipulator (ACM) under aerodynamic disturbances caused by a hovering unmanned aerial vehicle (UAV). The authors establish a nominal kinematic framework using strain-parameterized models and then estimate 3D position residuals using three neural architectures: a Multilayer Perceptron (MLP), a Gated Recurrent Unit (GRU), and a Closed-Form Continuous-Time (CfC) network. Experimental results demonstrate that the CfC network significantly outperforms the MLP and GRU, achieving a Root Mean Square Error (RMSE) of 22.00 mm compared to 36.38 mm and 27.72 mm, respectively, highlighting the effectiveness of continuous-time learning for handling aerodynamic disturbances.
Entities (7)
Relation Signals (5)
Closed-Form Continuous-Time Network → outperforms → Multilayer Perceptron
confidence 95% · the CfC achieves an RMSE of 22.00±1.70 mm ... compared with 36.38±3.58 mm for the MLP
Closed-Form Continuous-Time Network → outperforms → Gated Recurrent Unit
confidence 95% · the CfC achieves an RMSE of 22.00±1.70 mm ... compared with 27.72±2.92 mm for the GRU
Aerial Continuum Manipulator → affectedby → Aerodynamic Disturbance
confidence 94% · operating under aerodynamic effects induced by the unmanned aerial vehicle (UAV)
Closed-Form Continuous-Time Network → usedfor → Residual Estimation
confidence 93% · The selected nominal model then serves as the baseline for 3D position residual estimation using a closed-form continuous-time (CfC) neural network
Strain-Parameterized Kinematic Model → usedfor → Nominal Position Estimation
confidence 90% · To establish a nominal framework, strain-parameterized kinematic models ... are evaluated
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract: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.
Tags
Links
- Source: https://arxiv.org/abs/2609.28716v1
- Canonical: https://arxiv.org/abs/2609.28716v1
Trouble viewing inline? Open PDF directly →
Full Text
41,200 characters extracted from source content.
Expand or collapse full text
Temporal Learning for End-Effector Position Estimation under Aerodynamic Disturbances in Aerial Continuum Manipulation Niloufar Amiri Houman Masnavi Farrokh Janabi-Sharifi †thanks: This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant 2023-05542 and by the National Research Council Canada (NRC) under Grant AI4L-128-1.†thanks: Niloufar Amiri and Farrokh Janabi-Sharifi are with the Department of Mechanical, Industrial, and Mechatronics Engineering, Toronto Metropolitan University, Toronto, ON, Canada. Houman Masnavi is a Research Fellow at the University of Freiburg, Freiburg, Germany. E-mail: niloufar.amiri@torontomu.ca (corresponding author). Abstract This paper investigates temporal neural networks for end-effector position 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 (rotor-off) and free-hovering conditions across continuum robot (CR) configurations and UAV altitudes, providing end-effector position measurements with and without aerodynamic residuals. To establish a nominal framework, 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 closed-form 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±1.70m22.00± 1.70~m over five random seeds, compared with 36.38±3.58m36.38± 3.58~m for the MLP and 27.72±2.92m27.72± 2.92~m for the GRU, corresponding to reductions of 39.52%39.52\% and 20.62%20.62\%, respectively. These results demonstrate the effectiveness of continuous-time learning for end-effector position estimation under aerodynamic disturbances relative to static and discrete-time learning methods. Index Terms: Aerial manipulation, continuum robots, aerodynamic disturbance, residual estimation, temporal neural networks. I Introduction The integration of soft and continuum manipulators into aerial robotic systems enables enhanced dexterity in elevated and confined environments and potentially safer physical interaction with surrounding objects [1]. These advantages, however, introduce additional modeling and estimation challenges due to the compliant and nonlinear behavior of aerial continuum manipulators (ACMs) [2, 3]. In particular, their lightweight and deformable structures make them more susceptible than rigid aerial manipulators to aerodynamic disturbances induced by the unmanned aerial vehicle (UAV) [4, 5], leading to measurable deviations in the continuum robot (CR) configuration and end-effector position [6]. Accurate position estimation under these conditions requires both an appropriate nominal representation of the CR and a means of capturing the residual effects introduced by the induced airflow. Establishing the nominal response is itself nontrivial, since the accuracy of strain-parameterized continuum models depends on the spatial order used to approximate the strain distribution along the backbone. Increasing the strain basis order can capture more complex spatial variations, but also introduces additional generalized coordinates and model complexity [7, 8]. An appropriate nominal model should therefore provide sufficient accuracy without unnecessarily high-order parameterizations that yield only marginal improvement [9]. The end-effector position residual under aerodynamic disturbances presents an additional challenge because the downwash loading experienced by the CR varies with its configuration and UAV operating conditions, while the compliant response can depend on motion history through hysteresis and delayed effects. Consequently, the resulting position deviation may not be adequately represented by instantaneous system variables alone. Static neural models can approximate nonlinear input–output relationships but do not retain temporal information across sequential samples. Recurrent neural networks maintain a hidden memory of previous observations, while closed-form continuous-time networks additionally model the time-dependent evolution of this state, providing a representation more naturally aligned with continuously evolving physical dynamics [10, 11]. These characteristics motivate the investigation of temporal neural networks for end-effector position estimation under UAV-induced aerodynamic uncertainty in ACMs. I Related Work Disturbances in aerial manipulation are commonly treated as lumped uncertainties incorporating aerodynamic effects, platform vibration, modeling errors, and UAV–manipulator coupling. Model-based approaches, including nonlinear disturbance observers and adaptive methods, have been widely employed for their estimation and rejection [12, 13, 14]. However, these methods generally rely on sufficiently accurate system models and assumptions regarding the structure or bounds of unknown disturbances, which can be difficult to establish for highly coupled and nonlinear systems subject to complex aerodynamic interactions. Learning-based approaches have therefore been employed to approximate unmodeled dynamics and external disturbances using adaptive radial basis function (RBF) networks and deep neural networks [15, 16, 17, 18]. More specifically, aerodynamic disturbances have been investigated using analytical and data-driven approaches [19, 20, 21, 22, 23], although these studies primarily consider individual UAVs or aerodynamic interactions among multiple aerial vehicles. For ACMs, recent experiments have shown that rotor downwash can significantly affect CR deformation and end-effector position [6, 24]. In the most closely related study, Uthayasooriyan et al. [6] developed a Gaussian process regression model guided by constant curvature (C) kinematics for steady-state residual correction using a mechanically constrained multirotor. In contrast, this work considers a free-hovering UAV, establishes a strain-parameterized nominal model through a systematic study of spatial strain complexity, and estimates 3D position residuals using static, discrete-time recurrent, and continuous-time neural models across CR configurations and hovering altitudes. This formulation extends steady-state correction toward temporal residual estimation, explicitly examining the roles of temporal memory, continuous-time modeling, and nominal model complexity in uncertain position estimation. The main contributions are: • An experimental dataset characterizing ACM end-effector position under stationary and free-hovering UAV conditions across CR configurations, hovering altitudes, and UAV operating conditions. • A comparative evaluation of strain-parameterized kinematic models with increasing spatial strain basis order to establish a compact nominal representation for aerodynamic residual estimation. • A systematic comparison of MLP, GRU, and CfC networks for 3D end-effector position residual estimation under free-hovering UAV conditions, extending existing steady-state, C-based correction by evaluating temporal memory and continuous-time modeling. I Problem Formulation and Methodology I-A Nominal Strain-Parameterized Model Let s∈[0,L]s∈[0,L] denote the arc-length coordinate along a CR, where L is its undeformed length. The configuration of a cross-section at s relative to the base frame is represented by (s)=[(s)(s)1×31]∈SE(3), g(s)= bmatrix R(s)& r(s)\\ 0_1× 3&1 bmatrix (3), (1) where (s)∈SO(3) R(s) (3) and (s)∈ℝ3 r(s) ^3 denote the cross-section orientation and position, respectively. Following strain-parameterized Cosserat rod theory [8, 7], the configuration evolves as ′(s)=(s)^(s),^(s)=[S()1×30], g (s)= g(s) ξ(s), ξ(s)= bmatrixS( κ)& ν\\ 0_1× 3&0 bmatrix, (2) where S(⋅)S(·) is the skew-symmetric operator and =[⊤,⊤]⊤∈ℝ6 ξ=[ κ , ν ] ^6 is the local strain vector. Here, ∈ℝ3 κ ^3 contains the bending and torsional strains, while ∈ℝ3 ν ^3 contains the axial and shear strains. The distributed strain field is approximated by (s,s)=⋆+m(sL)s, ξ(s, q_s)= ξ + B_m\! ( sL ) q_s, (3) where ⋆ ξ is the reference strain, s∈ℝns q_s ^n_s contains the generalized strain coordinates, and m∈ℝ6×ns B_m ^6× n_s is the spatial strain basis of order m. The corresponding forward kinematics (FK) is (s,s)= g(s; q_s)= (0)exp(∫0s^(σ,s)dσ), g(0)\,P \! ( _0^s ξ(σ, q_s)\,dσ ), (4) where P denotes the path-ordering operator. The kinematics are numerically evaluated using Lie-group integration to preserve the SE(3)SE(3) structure [7]. For the tendon-driven CR considered in this work, shown in Fig. 1, the base frame is the attachment frame OaO_a, such that (0)=4 g(0)= I_4. The straight, unsheared reference strain is written as ⋆=[3⊤,b⊤]⊤ ξ =[ 0_3 , e_b ] , where b e_b is the unit vector along the undeformed backbone in the adopted local frame. Only the two actuated bending strains are parameterized, while torsion, extension, and shear remain fixed at their reference values. Shifted Legendre polynomials are employed as spatial basis functions. Constant, linear, quadratic, and cubic orders, corresponding to m=0,1,2,3m=0,1,2,3, are evaluated to examine the tradeoff between nominal prediction accuracy and spatial model complexity. Based on this comparison, the linear basis is selected for the nominal model. For the selected model, the two bending strain distributions are κ1(s) _1(s) =A1[1+a1P1(sL)], =A_1\! [1+a_1P_1\! ( sL ) ], (5) κ2(s) _2(s) =A2[1+a2P1(sL)], =A_2\! [1+a_2P_1\! ( sL ) ], where P1(η)=2η−1P_1(η)=2η-1 is the first-order shifted Legendre polynomial. Accordingly, s=[A1a1A1A2a2A2]⊤, q_s= bmatrixA_1&a_1A_1&A_2&a_2A_2 bmatrix , (6) where A1A_1 and A2A_2 are bending amplitudes, while a1a_1 and a2a_2 govern their spatial variation. The bending amplitudes are related to the measured actuator coordinates a=[q1,q2]⊤ q_a=[q_1,q_2] through a calibrated actuation mapping [A1A2]=(a,Φ), bmatrixA_1\\ A_2 bmatrix= ( q_a; θ_ ), (7) where (⋅) (·) denotes the actuator-to-strain mapping and Φ θ_ collects its calibration parameters. The mapping is calibrated separately from the spatial strain parameters. For the selected linear strain model, the remaining spatial parameters are collected as s=[a1,a2]⊤ θ_s=[a_1,a_2] . The nominal end-effector position is obtained from the translational component L=(L) r_L= r(L) of the tip configuration as ^nom=L+0, p^nom= C r_L+ p_0, (8) where ∈ℝ3×3 C ^3× 3 and 0∈ℝ3 p_0 ^3 denote the fixed coordinate registration and translation used to express the model prediction in the experimental measurement frame. After calibration of the actuator-to-strain mapping, the spatial parameters are identified from the stationary baseline measurements by minimizing the end-effector position error, s⋆=argmins∑i=1Nstat‖istat−^nom(a,i;s)‖22, θ_s = *arg\,min_ θ_s _i=1^N_stat \| p_i^stat- p^nom( q_a,i; θ_s) \|_2^2, (9) where NstatN_stat is the number of stationary measurements and istat∈ℝ3 p_i^stat ^3 is the corresponding measured end-effector position. The resulting nominal model is fixed before the subsequent aerodynamic residual learning process. Fig. 1: ACM frame attachment and operating conditions across different CR configurations. I-B UAV-Induced Residual Estimation The nominal model represents the CR response identified under stationary (rotor-off) conditions. Under free-hovering conditions, the measured end-effector position deviates from this nominal response due to UAV-induced aerodynamic effects and associated operating conditions. To estimate this deviation, the neural network input at sample k is defined as k=[q1,kq2,kq˙1,khkτk]⊤, u_k= bmatrixq_1,k&q_2,k& q_1,k&h_k& _k bmatrix , (10) where q1,kq_1,k and q2,kq_2,k are the actuator coordinates, q˙1,k q_1,k is the reciprocating actuation rate, hkh_k is the UAV hovering altitude, and τk _k is the recorded throttle signal. For the corresponding actuator configuration, the 3D position residual is defined as Δk=khov−^knom, p_k= p^hov_k- p^nom_k, (11) where khov∈ℝ3 p^hov_k ^3 is the measured CR end-effector position during free hovering and ^knom p^nom_k is the corresponding prediction of the nominal model. The residual is estimated as Δ^k=ℱ(k,k−1,…), p_k=F( u_k, u_k-1,…), (12) where the dependence on preceding observations is determined by the selected neural architecture. The corrected position estimate is then ^khov=^knom+Δ^k. p^hov_k= p^nom_k+ p_k. (13) Because Δk p_k is defined relative to an experimentally calibrated nominal model, it can contain both UAV-induced position deviations and residual nominal model mismatch. It is therefore interpreted as an effective UAV-induced position residual rather than a direct measurement or estimate of aerodynamic force. I-C Comparative Neural Architectures Three neural architectures with different temporal modeling capabilities are considered: an MLP, a GRU, and a CfC network. The MLP provides a memoryless baseline, the GRU introduces discrete-time recurrent memory, and the CfC employs a closed-form state transition derived from continuous-time neural dynamics [11]. For the recurrent models, consecutive measurements are arranged into finite input sequences. For a recurrent architecture with sequence length NsN_s, a sequence ending at sample k is defined as k(Ns)=[k−Ns+1,…,k]. U_k^(N_s)= [ u_k-N_s+1,…, u_k ]. (14) The sequence length NsN_s is selected separately for the GRU and CfC architectures. Sequences are constructed independently within each experiment, and a temporal gap exceeding a prescribed threshold Δtmax t_ initiates a new sequence segment. The MLP, GRU, and CfC models are evaluated at identical target timestamps to ensure a direct comparison. I-C1 Multilayer Perceptron The MLP estimates the residual from the instantaneous input vector. For hidden layer ℓ , k(ℓ)=ρ((ℓ)k(ℓ−1)+(ℓ)),k(0)=k, a_k^( )=ρ ( W^( ) a_k^( -1)+ b^( ) ), a_k^(0)= u_k, (15) and the output is Δ^k=oMLPk(Lm)+oMLP. p_k= W_o^MLP a_k^(L_m)+ b_o^MLP. (16) Here, (ℓ) W^( ) and (ℓ) b^( ) denote the trainable weight matrix and bias vector of hidden layer ℓ , respectively, while oMLP W_o^MLP and oMLP b_o^MLP denote the output-layer parameters. The function ρ(⋅)ρ(·) denotes the nonlinear activation function, and LmL_m is the number of hidden layers. The MLP therefore implements the static mapping Δ^k=ℱMLP(k), p_k=F_MLP( u_k), (17) without an explicit recurrent state. I-C2 Gated Recurrent Unit The GRU introduces a recurrent hidden state k h_k to retain information from preceding observations [25]. Its reset and update gates are defined as k r_k =σ(irk+ir+hrk−1+hr), =σ ( W_ir u_k+ b_ir+ W_hr h_k-1+ b_hr ), (18) k z_k =σ(izk+iz+hzk−1+hz), =σ ( W_iz u_k+ b_iz+ W_hz h_k-1+ b_hz ), (19) with candidate hidden state ~k=tanh(ink+in+k⊙(hnk−1+hn)). h_k= ( W_in u_k+ b_in+ r_k ( W_hn h_k-1+ b_hn) ). (20) The recurrent state is updated according to k=(−k)⊙~k+k⊙k−1, h_k=( 1- z_k) h_k+ z_k h_k-1, (21) and mapped to the residual estimate through Δ^k=oGRUk+oGRU. p_k= W_o^GRU h_k+ b_o^GRU. (22) Here, i(⋅) W_i(·) and h(⋅) W_h(·) denote the trainable input and recurrent weight matrices, respectively, and i(⋅) b_i(·) and h(⋅) b_h(·) are the corresponding bias vectors. The parameters oGRU W_o^GRU and oGRU b_o^GRU define the output mapping. The function σ(⋅)σ(·) denotes the sigmoid function and ⊙ is the Hadamard product. I-C3 Closed-Form Continuous-Time Network The CfC architecture employs a closed-form recurrent state transition derived from continuous-time neural dynamics, avoiding numerical integration of the underlying state evolution [11]. Defining the augmented input to the recurrent transition as k=[k⊤k−1⊤]⊤. η_k= bmatrix u_k & h_k-1 bmatrix . (23) The elapsed time between consecutive measurements is defined as Δtk=tk−tk−1 t_k=t_k-t_k-1, where tkt_k denotes the timestamp associated with sample k. A time-dependent interpolation gate is then expressed as k=σ(a(k)Δtk+b(k)), α_k=σ (N_a( η_k) t_k+N_b( η_k) ), (24) and the recurrent state is updated according to k= h_k= (−k)⊙1(k)+k⊙2(k), ( 1- α_k) _1( η_k)+ α_k _2( η_k), (25) where 1N_1, 2N_2, aN_a, and bN_b denote trainable nonlinear mappings. The dependence on Δtk t_k allows the recurrent transition to account explicitly for variations in the time interval between consecutive measurements. The residual estimate is obtained as Δ^k=oCfCk+oCfC. p_k= W_o^CfC h_k+ b_o^CfC. (26) IV Experimental Setup and Dataset Generation Fig. 1 shows the experimental platform used for dataset generation, consisting of a tendon-driven CR rigidly mounted on a UAV. The CR workspace is partitioned into 12 slices defined by prescribed ranges of q1q_1. For each experiment, q1q_1 is set to the midpoint of the corresponding range and held constant, while q2q_2 undergoes reciprocating motion. Thus, q˙1=0 q_1=0 during each experiment and is not included as a model input. By symmetry, each experiment represents four workspace slices; consequently, three experiments cover all 12 slices at a given hovering altitude. The experiments are repeated at four hovering altitudes, resulting in 12 free-hovering experiments. Five additional baseline experiments are conducted with inactive rotors to characterize the nominal CR behavior without UAV-induced airflow. During each experiment, the CR is actuated sufficiently slowly such that inertial effects are negligible and its motion can be treated as kinetostatic. The 3D positions of the CR end-effector and UAV center of mass are measured at 10Hz10~Hz using a Vicon motion capture system (Vicon Motion Systems Ltd., UK), while UAV throttle and the two tendon-actuating servo encoder measurements are recorded. The complete dataset contains 28,411 samples, including 10,538 baseline samples and 17,873 samples collected under UAV operation. For residual estimation, Tests 13–16 (with payload) are excluded, and domain filtering of Tests 1–12 retains 13,922 free-hovering samples. The data are split at the experiment level, with Tests 1–9 forming the development dataset, Tests 10–12 reserved as the unseen dataset for final evaluation, and Tests 17–21 used exclusively for nominal model calibration. V Results and Analysis V-A Nominal Model Evaluation The nominal CR kinematics is identified from the baseline dataset in two steps. First, strain-parameterized models with constant through cubic spatial bases are compared using Test 17 (q2=0q_2=0), where deformation occurs predominantly in the principal bending plane. Measurements with similar q1q_1 values are averaged to approximate the nominal static relationship and reduce variability due to reciprocating motion. As shown in Fig. 2(a) and (b), the constant strain model yields a binned x–z RMSE of 94.72m94.72~m, whereas the linear basis reduces the error to 13.84m13.84~m, an 85.4%85.4\% reduction. Quadratic and cubic bases provide only marginal further improvement, reaching approximately 13.03m13.03~m. The linear basis is therefore selected as a compact representation of the dominant nonuniform deformation. These results indicate that the C assumption, equivalent to constant strain in the present formulation, poorly represents the actual CR deformation. The selected strain model is then evaluated over the complete baseline dataset against a purely data-driven third-order polynomial FK . As shown in Fig. 2(c)–(g), the two models achieve comparable accuracy across the workspace. The polynomial model yields RMSEs of (16.13,,7.53,,17.37)m(16.13,,7.53,,17.37)~m in x, y, and z, respectively, with a 3D RMSE of 24.87m24.87~m, compared with (16.94,,8.48,,16.60)m(16.94,,8.48,,16.60)~m and 25.19m25.19~m for the strain model. Despite similar accuracy, the strain model and actuation mapping together use 20 identified parameters, compared with 30 coefficients in the purely data-driven polynomial model, while preserving a physically structured representation of the distributed CR deformation rather than directly fitting the Cartesian tip coordinates. Fig. 2: Nominal model evaluation. (a) x–z RMSE of strain models with different basis orders on Test 17 using binned and raw data. (b) RMSE of the third-order polynomial and selected linear strain models over the complete baseline dataset. (c) Per-test RMSE for Tests 17–21. (d)–(f) Measured and predicted xtipx_tip, ytipy_tip, and ztipz_tip versus centered q1∗q_1^* across the five baseline tests; colors distinguish tests, and solid/dashed curves denote polynomial/strain models. (g)–(i) Test 17 x–z trajectories comparing measured data and the constant strain model with the linear, quadratic, and cubic strain models, respectively. V-B UAV-Induced Residual Characterization Before evaluating the residual estimators, the UAV-induced position residual is characterized across the CR workspace and hovering conditions to quantify its 3D magnitude and dependence on CR coordinates and UAV altitude. Across Tests 1–12, the residual has an overall 3D RMS of 44.05m44.05~m, as shown in Fig. 3(a). The residual varies with both flight altitude and CR configuration. Fig. 3(b) and (c) show a nonmonotonic dependence on altitude, while Fig. 3(d)–(f) reveal nonlinear variations with q1∗q_1^* that also differ across the q2q_2 configuration slices. Residuals are present in all three Cartesian directions, with the largest RMS component along z, consistent with the primary direction of the rotor-induced downwash acting on the CR beneath the UAV. Fig. 3: Aerodynamic residual characterization for free-hovering Tests 1–12. (a) Overall RMS of the Cartesian residual components Δx x, Δy y, and Δz z, and the 3D residual. (b) Directional residual RMS at four flight altitudes h1h_1–h4h_4, ordered by increasing mean altitude. (c) 3D residual RMS versus mean flight altitude. (d)–(f) Mean Δx x, Δy y, and Δz z versus centered q1∗q_1^* for the three configuration slices q2≈0q_2≈ 0, q2≈1.2q_2≈ 1.2, and q2≈2.8q_2≈ 2.8. Solid curves show binned mean values, with shaded regions indicating ±1± 1 standard deviation. V-C Learning-Based Residual Estimation The development dataset is utilized to train and compare the MLP, GRU, and CfC estimators for 3D UAV-induced residual prediction. Hyperparameters are selected using three-fold cross-validation, with each fold using one experiment from each development altitude for validation and the remaining experiments for training. Thus, all development altitudes are represented in both partitions. Input and target standardization statistics are computed exclusively from each training partition and the unseen dataset remains reserved for final evaluation. Cross-validation selects sequence lengths of 25 and 50 samples for the GRU and CfC, respectively, with 32 recurrent units and a learning rate of 10−310^-3 for both models. The MLP contains two hidden layers of 64 ReLU units each. All models are trained using the Adam optimizer and mean squared error (MSE) loss. Final performance is evaluated on the unseen dataset, collected at a hovering altitude excluded from neural network development, with the nominal linear strain-parameterized model used as the uncompensated reference. Fig. 4 summarizes the prediction performance and shows the estimated residuals and instantaneous position errors across the unseen experiments. As shown in Fig. 4(a), the nominal model yields a 3D RMSE of 44.94m44.94~m, which is reduced to 41.98m41.98~m by the MLP. The temporal models provide substantially larger improvements, reducing the 3D RMSE to 27.75m27.75~m for the GRU and 21.87m21.87~m for the CfC. The directional errors show a similar trend, with particularly notable improvements in the local x and z directions. The residual R2R^2 values in Fig. 4(b) further demonstrate the benefit of temporal modeling. The MLP yields negative R2R^2 values for Δx x and Δz z, indicating poor generalization of the memoryless mapping for these residual components. In contrast, the GRU achieves positive R2R^2 values in all three directions, while the CfC reaches approximately 0.750.75, 0.390.39, and 0.670.67 for Δx x, Δy y, and Δz z, respectively. The distribution of true versus predicted values in Fig. 4(d)–(f) similarly shows that the CfC captures the dominant residual trends, although greater dispersion remains in the y direction, which may reflect the more limited workspace variation along this dimension. The time histories in Fig. 4(g)–(i) further show that the temporal models generally maintain lower instantaneous 3D position errors across the unseen experiments, although performance varies with operating conditions and Test 12 remains the most challenging case. These results show that incorporating temporal information substantially improves residual estimation compared with a memoryless nonlinear mapping for the present ACM system. Fig. 4: Residual estimators performance on unseen dataset. V-D Model Sensitivity and Robustness Input ablation and robustness analyses are conducted to evaluate the contribution of each input to prediction accuracy and the sensitivity of the learned estimators to stochastic training. For the input ablation, the selected temporal models are retrained by individually removing q˙1 q_1, hovering altitude, and rotor throttle while keeping the remaining model settings fixed. The same three-fold cross-validation protocol used during model development is applied to each ablation case to ensure a fair comparison. Robustness is evaluated by independently training each final model configuration with five random seeds and evaluating each model on the same unseen dataset, without further hyperparameter adjustment. Table I summarizes the input ablation results using the same cross-validation employed during model development. Removing the actuation rate q˙1 q_1 increases the mean 3D RMSE by 11.34%11.34\% for the CfC and 11.20%11.20\% for the GRU. This consistent degradation indicates that rate information captures recent CR motion and delayed residual effects beyond the instantaneous configuration, supporting its inclusion in the estimator input. Hovering altitude also contributes to prediction accuracy. Removing altitude increases the 3D RMSE by 18.89%18.89\% for the CfC and 7.52%7.52\% for the GRU. For the CfC, the largest directional degradation occurs in the y direction, where the RMSE increases from 18.2718.27 to 28.00m28.00~m. This behavior is consistent with the residual characterization, which shows that the aerodynamic residual varies with hovering altitude without following a simple monotonic trend. In contrast, removing rotor throttle slightly reduces the validation RMSE by 2.05%2.05\% for the CfC and 3.60%3.60\% for the GRU. Although rotor thrust directly influences the generated downwash, the UAV operated near its available thrust capacity because of the limited payload margin, resulting in a relatively narrow throttle range during the experiments. Consequently, the throttle signal provided limited additional predictive information in the present dataset. TABLE I: Input ablation analysis for the temporal estimators. The reported ΔRMSE is relative to the corresponding full-input model, with positive values indicating increased error. Model Removed 3D RMSE [m] ΔRMSE [%] x [m] y [m] z [m] CfC None 29.62±11.5929.62± 11.59 – 17.48 18.27 13.47 q˙1 q_1 32.97±10.8632.97± 10.86 +11.34+11.34 22.68 15.91 16.84 Altitude 35.21±7.0335.21± 7.03 +18.89+18.89 16.63 28.00 11.37 Throttle 29.01±11.3629.01± 11.36 −2.05-2.05 16.36 17.76 14.18 GRU None 31.44±10.3531.44± 10.35 – 19.43 17.04 16.07 q˙1 q_1 34.96±9.7334.96± 9.73 +11.20+11.20 22.91 17.54 17.75 Altitude 33.80±7.6733.80± 7.67 +7.52+7.52 16.78 25.45 12.21 Throttle 30.31±11.2730.31± 11.27 −3.60-3.60 17.80 18.83 14.68 TABLE I: Robustness analysis on the unseen dataset over five random seeds. Improvement is measured relative to the nominal linear strain FK. Model x [m] y [m] z [m] 3D RMSE [m] Improvement [%] Linear strain FK 25.54 19.94 31.14 44.94 – MLP 24.38±2.0524.38± 2.05 13.26±2.6813.26± 2.68 23.24±4.2323.24± 4.23 36.38±3.5836.38± 3.58 19.04±7.9819.04± 7.98 GRU 15.58±2.2215.58± 2.22 16.03±1.3016.03± 1.30 16.31±2.2316.31± 2.23 27.72±2.9227.72± 2.92 38.32±6.5038.32± 6.50 CfC 11.92±0.8511.92± 0.85 14.27±1.8614.27± 1.86 11.72±0.3911.72± 0.39 22.00±1.7022.00± 1.70 51.04±3.7851.04± 3.78 Table I presents the robustness analysis on the unseen experiments. The nominal linear strain FK yields a 3D RMSE of 44.94m44.94~m. Across five random seeds, the MLP achieves 36.38±3.58m36.38± 3.58~m, corresponding to a 19.04±7.98%19.04± 7.98\% improvement relative to the nominal model. The GRU reduces the error to 27.72±2.92m27.72± 2.92~m, corresponding to an improvement of 38.32±6.50%38.32± 6.50\%, while the CfC achieves 22.00±1.70m22.00± 1.70~m and an improvement of 51.04±3.78%51.04± 3.78\%. In terms of mean 3D RMSE, the CfC therefore reduces the error by approximately 39.5%39.5\% relative to the MLP and 20.6%20.6\% relative to the GRU. It also exhibits the smallest variation in 3D RMSE across the five random seeds and achieves the lowest 3D error in each run. The component-wise results show that the CfC achieves the lowest RMSE in the x and z directions, with 11.92±0.85m11.92± 0.85~m and 11.72±0.39m11.72± 0.39~m, respectively. Although the MLP yields the lowest mean y-direction RMSE, its larger errors in the other directions result in a substantially higher overall 3D RMSE, highlighting the importance of evaluating the complete 3D position error rather than individual Cartesian components. Performance also varies across the individual unseen experiments, with Test 12 representing the most challenging condition for all three learned estimators. This test exhibits relatively larger lateral CR deviation at close ground proximity. The CfC yields lower errors than the GRU for Tests 11 and 12, whereas the two temporal models perform similarly on Test 10, for which the GRU achieves a slightly lower mean error. Thus, the CfC does not consistently outperform the GRU under every individual operating condition, but provides lower overall error and greater consistency across the complete unseen dataset. Overall, the robustness analysis reinforces the benefit of temporal modeling for UAV-induced residual estimation. Both temporal models achieve substantially lower 3D RMSE than the memoryless MLP, while the CfC provides the lowest mean 3D RMSE and smallest variation under the experimental conditions considered. VI Conclusion Two fundamental challenges in ACMs are investigated in this paper. These platforms are complex coupled systems with multibody and continuum dynamics, making accurate modeling challenging. Reduced-order models are therefore attractive for real-time deployment, although the trade-off between modeling accuracy and complexity must be carefully considered. Using an experimental dataset, a compact physics-informed strain model is evaluated and shown to provide accuracy comparable to that of a data-driven polynomial model while preserving a more structured representation. Real-world experiments also reveal noticeable deviations in CR end-effector position under UAV-induced aerodynamic effects. The effectiveness of learning-based residual estimation is therefore investigated, with temporal models providing substantially better performance than a memoryless approach on unseen experiments. These results demonstrate a lightweight and deployable approach for improving end-effector position estimation in ACMs. Future work will focus on extending the framework to more aggressive UAV maneuvers and stochastic aerodynamic conditions. References [1] A. Jalali and F. Janabi-Sharifi (2022) Aerial continuum manipulation: a new platform for compliant aerial manipulation. Frontiers in Robotics and AI 9, p. 903877. Cited by: §I. [2] N. Amiri and F. Janabi-Sharifi (2025) High-performance coupled kinematics of aerial continuum manipulation systems for control applications. Robotics and Autonomous Systems 192, p. 105021. Cited by: §I. [3] N. Amiri, S. Sepahvand, I. Mantegh, and F. Janabi-Sharifi (2026) Systematic analysis of coupling effects on closed-loop and open-loop performance in aerial continuum manipulators. In 2026 International Conference on Unmanned Aircraft Systems (ICUAS), Vol. , p. 1184–1191. External Links: Document Cited by: §I. [4] H. B. Khamseh, F. Janabi-Sharifi, and A. Abdessameud (2018) Aerial manipulation—a literature survey. Robotics and Autonomous Systems 107, p. 221–235. Cited by: §I. [5] A. Ollero, M. Tognon, A. Suarez, D. Lee, and A. Franchi (2021) Past, present, and future of aerial robotic manipulators. IEEE Transactions on Robotics 38 (1), p. 626–645. Cited by: §I. [6] A. Uthayasooriyan, K. M. Digumarti, F. Vanegas, and F. Gonzalez (2026) An experimental study of downwash effects on a continuum manipulator integrated with a multirotor uav. IEEE Robotics and Automation Letters. Cited by: §I, §I. [7] F. Renda, C. Armanini, V. Lebastard, F. Candelier, and F. Boyer (2020) A geometric variable-strain approach for static modeling of soft manipulators with tendon and fluidic actuation. IEEE Robotics and Automation Letters 5 (3), p. 4006–4013. Cited by: §I, §I-A, §I-A. [8] F. Boyer, V. Lebastard, F. Candelier, and F. Renda (2020) Dynamics of continuum and soft robots: a strain parameterization based approach. IEEE transactions on robotics 37 (3), p. 847–863. Cited by: §I, §I-A. [9] N. Amiri and F. Janabi-Sharifi (2026) Strain-parameterized coupled dynamics and dual-camera visual servoing for aerial continuum manipulators. arXiv preprint arXiv:2603.23333. Cited by: §I. [10] Z. C. Lipton, J. Berkowitz, and C. Elkan (2015) A critical review of recurrent neural networks for sequence learning. arXiv preprint arXiv:1506.00019. Cited by: §I. [11] R. Hasani, M. Lechner, A. Amini, L. Liebenwein, A. Ray, M. Tschaikowski, G. Teschl, and D. Rus (2022) Closed-form continuous-time neural networks. Nature Machine Intelligence 4, p. 992–1003. External Links: Document Cited by: §I, §I-C3, §I-C. [12] J. Liang, Y. Chen, Y. Wu, Z. Miao, H. Zhang, and Y. Wang (2022) Adaptive prescribed performance control of unmanned aerial manipulator with disturbances. IEEE Transactions on Automation Science and Engineering 20 (3), p. 1804–1814. Cited by: §I. [13] Y. Chen, J. Liang, Y. Wu, Z. Miao, H. Zhang, and Y. Wang (2022) Adaptive sliding-mode disturbance observer-based finite-time control for unmanned aerial manipulator with prescribed performance. IEEE transactions on cybernetics 53 (5), p. 3263–3276. Cited by: §I. [14] X. Liang, Y. Wang, H. Yu, Z. Zhang, J. Han, and Y. Fang (2024) Observer-based nonlinear control for dual-arm aerial manipulator systems suffering from uncertain center of mass. IEEE Transactions on Automation Science and Engineering 22, p. 1984–1995. Cited by: §I. [15] Q. Fang, P. Mao, L. Shen, and J. Wang (2023) Robust control based on adaptive neural network for the process of steady formation of continuous contact force in unmanned aerial manipulator. Sensors 23 (2), p. 989. Cited by: §I. [16] H. Li, Z. Li, J. Liu, X. Zheng, X. Yu, and O. Kaynak (2024) Adaptive neural network backstepping control method for aerial manipulator based on coupling disturbance compensation. Journal of the Franklin Institute 361 (7), p. 106733. Cited by: §I. [17] Y. Wu, Z. Zhou, M. Wei, and H. Cheng (2024) Robust and energy-efficient control for multi-task aerial manipulation with automatic arm-switching. In 2024 IEEE International Conference on Robotics and Automation (ICRA), p. 8394–8400. Cited by: §I. [18] M. Wang, Z. Chen, K. Guo, X. Yu, Y. Zhang, L. Guo, and W. Wang (2023) Millimeter-level pick and peg-in-hole task achieved by aerial manipulator. IEEE Transactions on Robotics 40, p. 1242–1260. Cited by: §I. [19] L. Bauersfeld, K. Muller, D. Ziegler, F. Coletti, and D. Scaramuzza (2024) Robotics meets fluid dynamics: a characterization of the induced airflow below a quadrotor as a turbulent jet. IEEE Robotics and Automation Letters 10 (2), p. 1241–1248. Cited by: §I. [20] C. Chen and H. H. Liu (2021) Adaptive modeling for downwash effects in multi-uav path planning. Guidance, Navigation and Control 1 (04), p. 2140005. Cited by: §I. [21] B. Wang, Z. Ma, S. Lai, and L. Zhao (2023) Neural moving horizon estimation for robust flight control. IEEE Transactions on Robotics 40, p. 639–659. Cited by: §I. [22] J. Li, L. Han, H. Yu, Y. Lin, Q. Li, and Z. Ren (2023) Nonlinear mpc for quadrotors in close-proximity flight with neural network downwash prediction. In 2023 62nd IEEE Conference on Decision and Control (CDC), p. 2122–2128. Cited by: §I. [23] P. Kharitenko, Y. Fan, X. Liu, and Y. Wang (2025) A spatiotemporal downwash modeling for agile close-proximity multirotor flight. In 2025 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), p. 807–812. Cited by: §I. [24] Y. Zhang, H. Song, Z. Hu, D. Wei, and W. Wang (2026) Motion control and experimental verification of a continuum aerial manipulator for power grid maintenance operations. Journal of Field Robotics, p. 1–18. External Links: Document Cited by: §I. [25] K. Cho, B. van Merrienboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio (2014) Learning phrase representations using rnn encoder–decoder for statistical machine translation. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), p. 1724–1734. Cited by: §I-C2.