# Hybrid Supervised Learning and Constrained Optimization for Task-Space Trajectory Generation in Tendon-Driven Continuum Robots

Mohammad Jabari, Carmen Visconte, Giuseppe Quaglia, Med Amine Laribi

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Mohammad Jabari, Carmen Visconte, Giuseppe Quaglia, Med Amine Laribi. Hybrid Supervised Learning and Constrained Optimization for Task-Space Trajectory Generation in Tendon-Driven Continuum Robots. ASME Open Journal of Engineering, 2026, 5, pp.051003. 10.1115/1.4070630. hal-05455065

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Mohammad Jabari<fnref n="1" />

Department of Informatics, Bioengineering, Robotics and Systems Engineering (DIBRIS), Università di Genova, Genova 16145, Italy; Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: mohammad.jabari@polito.it

Carmen Visconte

Associate Professor
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
10129 Torino, Italy
e-mail: carmen.visconte@polito.it

Giuseppe Quaglia

Full Professor
Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
10129 Torino, Italy
e-mail: giuseppe.quaglia@polito.it

Med Amine Laribi

Professor
Department of GMSC,
Prime Institute, University of Poitiers, CNRS,
ISEA-ENSMA, UPR 3346,
Poitiers 86962, France
e-mail: med.amine.laribi@univ-poitiers.fr

Hybrid Supervised Learning and
Constrained Optimization for
Task-Space Trajectory
Generation in Tendon-Driven
Continuum Robots

Tendon-driven continuum robots (TDCRs) offer exceptional dexterity and compliance, making them suitable for manipulation in constrained and complex environments. However, solving the inverse-kinematics (IK) problem with sufficient accuracy and physical feasibility remains challenging due to nonlinear curvature coupling and actuation uncertainties. This paper presents a hybrid inverse-kinematics framework that integrates supervised deep learning with constrained optimization to achieve precise and physically consistent task-space trajectory tracking. Neural networks provide fast initial pose-to-configuration estimates, which are subsequently refined through constraint-aware optimization, ensuring geometric and actuation feasibility. The training data were generated from optimal IK solutions obtained via constrained optimization across sixteen trajectory-constraint combinations, covering four representative trajectories circular, elliptical, helical, and butterfly, and four end-effector orientation modes. Extensive simulations on a two-segment tendon-driven continuum robot demonstrated statistically significant improvements ($p \le 0.01$) in both position and orientation accuracy compared with standalone neural or optimization-based approaches. The hybrid method achieves micrometer-level positional accuracy and micro-degree-level orientation precision while maintaining computational efficiency suitable for real-time applications. The present work focuses on a quasi-static modeling and trajectory-generation framework. Future work will extend constraint handling to include dynamic obstacle avoidance and smoothness optimization, explore global solvers such as genetic algorithms and particle swarm optimization, and address sim-to-real transfer through adaptive learning for multi-segment robots in realistic environments. [DOI: 10.1115/1.4070630]

Keywords: tendon-driven continuum robots, hybrid inverse kinematics, constrained optimization, deep learning, trajectory tracking, statistical validation, robot kinematics, robotics, robotics and flexible tooling

# 1 Introduction

Tendon-driven continuum robots (TDCRs) have gained increasing attention over the past few decades, as a fundamentally new robotic manipulation platform with the potential to bring compliance, high dexterity, and access to confined and unstructured workspaces to a variety of tasks and applications [1,2]. These features make them ideal for applications such as minimally invasive surgery [3,4], industrial inspection [5], and search and rescue [6]. Unlike rigid-link manipulators, TDCRs generate motion through deformation of their continuous or semi-continuous backbones, actuated by tensioned tendons [7]. While this morphology provides

exceptional flexibility, it introduces significant challenges in modeling, motion planning, and control, particularly for accurate end-effector trajectory generation [8,9]. The inherently high degrees-of-freedom (DoF) and nonlinear mapping between actuation and shape make it difficult to apply traditional robotics algorithms effectively [10–13].

Accurate and efficient kinematic modeling is critical for model-based planning and control of TDCRs. Among available approaches, the piecewise constant-curvature (PCC) model is widely used for its simplicity and effectiveness. It approximates the backbone as a series of constant-curvature arcs, reducing the dimensionality of the control problem by representing each segment with a compact set of parameters: curvature, bending plane, and arc length [14–18]. The PCC model has been successfully applied in forward and inverse kinematics [19,20], Jacobian-based control [21], and dynamic modeling [22]. However, its

<footnote n="1">¹Corresponding author.
Manuscript received August 10, 2025; final manuscript received December 8, 2025; published online January 12, 2026. Assoc. Editor: Sze Zheng Yong.</footnote>
simplifying assumptions can reduce accuracy under high loads, rapid motions, or material nonlinearities, necessitating thorough validation [23–25]. While more accurate alternatives like Cosserat-rod models exist, their computational cost limits real-time application [26–28]. Moreover, material properties significantly influence both modeling fidelity and robot performance [29].

Trajectory generation for TDCRs is further complicated by the high-dimensional, nonlinear configuration space. Classical approaches rely on model-based inverse kinematics and trajectory planning, often using the PCC model [30]. However, these methods struggle with singularities, convergence issues [31], and the inability to enforce essential constraints, such as tendon tension limits, actuator bounds, workspace restrictions, and obstacle avoidance [32,33]. Sensor integration for robust proprioception and environmental awareness also remains an ongoing challenge [34]. To address these limitations, optimization-based strategies formulate trajectory generation as a constrained problem that incorporates physical and task-level constraints. Yet, they typically require high-quality initial guesses and are computationally intensive, especially for high-DoF systems [33–37].

Recently, learning-based methods, particularly supervised deep learning, have gained traction for modeling inverse kinematics due to their ability to approximate complex mappings in high-dimensional spaces without explicit models [38,39]. Applied to TDCRs, these methods can map task-space trajectories to actuation parameters, enabling efficient tracking while handling model uncertainties [39,40]. However, they rely heavily on high-quality, often simulated training data, leading to sim-to-real transfer challenges [41,42]. Furthermore, they lack built-in mechanisms to enforce physical constraints or guarantee feasibility [43,44], and designing loss functions that ensure smooth, accurate, and constrained behavior is non-trivial. Training quality also depends on the fidelity of the forward model used to generate data, highlighting the need for accurate parameter identification strategies [45].

To overcome the limitations of standalone model-based or data-
driven methods, hybrid approaches that combine supervised learn-
ing with constrained optimization have emerged as a promising
solution [44]. Neural networks offer fast initial estimates for
inverse kinematics, while optimization ensures adherence to phys-
ical and safety constraints. In TDCRs, this two-stage hybrid
process learning-based estimation followed by physics-aware
refinement facilitates feasible, accurate trajectory generation. It
also enhances safety, robustness under uncertainty, and applicabil-
ity to collaborative scenarios. However, maintaining system stabi-
lity and reliability remains an open research challenge [45].

While several hybrid inverse-kinematics (IK) and trajectory-
generation frameworks have been proposed for tendon-driven con-
tinuum robots based on either PCC or Cosserat-rod formulations,
most existing approaches face a tradeoff between accuracy and
real-time feasibility. PCC-based hybrids typically achieve faster
computation but suffer from reduced shape fidelity, whereas
Cosserat-rod formulations provide higher accuracy at the cost of
heavy computation and poor scalability [46,47]. The present frame-
work combines a data-driven estimator with constraint-aware opti-
mization, preserving the analytical simplicity of PCC while
improving convergence robustness and numerical stability for real-
time deployment.

A concise comparative overview is provided in Table 1, posi-
tioning the proposed method with respect to representative
hybrid (IK) and trajectory-generation approaches. The results
emphasize that this work achieves the best balance among real-
time performance, constraint handling, and model simplicity.

This paper presents a hybrid trajectory-generation framework for TDCRs that integrates supervised deep learning with constrained optimization, grounded in the PCC model. The PCC formulation is used both to generate training data and to define the optimization objectives and constraints. We hypothesize that a neural network can approximate the mapping from task-space poses to curvature parameters, and that subsequent constrained optimization can refine these estimates to ensure feasibility and

accuracy. Simulations on a two-segment tendon-driven continuum robot (TDCR) demonstrate that our hybrid approach outperforms standalone learning- and optimization-based methods in terms of accuracy, constraint satisfaction, and computational efficiency.

The main contributions of this work are as follows:

* A hybrid trajectory-generation framework combining supervised learning and constrained optimization for TDCRs, based on the PCC model.

* A deep neural network (DNN) trained to estimate curvature parameters from task-space poses, enabling fast inverse kinematic model initialization.

* A constrained optimization formulation that refines predictions by enforcing physical feasibility and trajectory accuracy.

* Simulation-based validation on a two-segment TDCR, showing improved performance over baseline approaches.

Compared with previously reported hybrid approaches that either integrate model-based optimization with limited data-driven correction or employ deep reinforcement strategies for adaptive learning, the present study adopts a distinct data-driven-optimization integration pathway.

Specifically, the proposed framework generates its training data not from random sampling or forward simulation alone, but from the optimal solutions of the inverse-kinematics problem under multiple task conditions.

These data were systematically obtained by solving the con-
strained IK optimization for four representative trajectories (circular, elliptical, helical, and butterfly) and four distinct end-effector orientation modes (Fixed Tilt around Y, Lock X-axis, Lock Y-axis, and Lock Z-axis), yielding sixteen trajectory constraint configurations.

This strategy ensures that the neural networks (MLP, DNN, and HybNN) learn from physically feasible and kinematically consistent mappings derived from optimal IK solutions, rather than approximating arbitrary or noisy configurations.

Consequently, the proposed method establishes a technically distinct hybrid paradigm combining data generated from physics-consistent optimization with constraint-aware neural inference that enhances accuracy, reproducibility, and generalization compared with conventional hybrid or reinforcement-based approaches.

The remainder of this article is structured as follows: Sec. 2 details the PCC-based kinematic model with vertebra-level parameterization and tendon-actuated constraint formulation. Section 3 introduces the proposed hybrid (IK) framework, integrating supervised learning with constrained optimization. Section 4 outlines the simulation setup, dataset generation, and neural network training process. Section 5 reports the evaluation results, including trajectory tracking accuracy, tendon actuation profiles, and comparative error analysis across trajectories and constraints. Section 6 concludes the paper with a summary of findings and directions for future research.

## 2 Modeling Framework for a Two-Segment Continuum Robot

This section presents a comprehensive kinematic model for a
tendon-driven continuum robot composed of two serially con-
connected segments, each actuated by four symmetrically routed
tendons and consisting of multiple rigid vertebrae connected by
compliant joints that enable smooth, continuous deformation.
The global configuration of the robot, along with the coordinate
conventions used throughout this study, is illustrated in Fig. 1,
where each segment is represented as a smooth arc defined by its
overall bending angle $\theta_i$ and bending-plane orientation $\phi_i$.

The modeling framework builds upon the classical Piecewise
Constant-Curvature (PCC) formulation, which approximates each
segment as a planar arc with uniform curvature. While effective
for many applications, this assumption restricts motion to planar
<table><caption>Table 1 Comparative summary of hybrid (IK) and trajectory-generation frameworks¹</caption><thead><tr><th>Method/Reference</th><th>Model basis</th><th>Constraint handling</th><th>Real-time capability</th><th>Validation</th><th>Main limitation</th></tr></thead><tbody><tr><td>[20]</td><td>PCC</td><td>Partial (curvature)</td><td>X</td><td>Simulation</td><td>Slow convergence</td></tr><tr><td>[22]</td><td>Cosserat-rod</td><td>Full (geometric)</td><td>X</td><td>Simulation</td><td>High computation cost</td></tr><tr><td>[45]</td><td>RL + Kinematic</td><td>Implicit</td><td>Δ</td><td>Simulation</td><td>Requires large data</td></tr><tr><td>This work</td><td>PCC + NN + Constrained Opt.</td><td>Full (pose + tendon)</td><td>✓ (&gt;7 Hz)<sup>2</sup></td><td>✓ (Simulation)</td><td>Quasi-static assumption</td></tr></tbody></table>

Note: ✓, full support; Δ, partial or implicit capability; X, not supported.

bending and neglects torsional deformation. To overcome this limitation, the proposed formulation introduces a per-vertebra extension that incorporates axial twist in addition to planar bending.

Each vertebra is modeled as a discrete subsegment capable of independent bending and twisting, allowing the representation of complex, spatially non-planar configurations beyond the descriptive capacity of the conventional PCC model.

Specifically, each vertebra is characterized by three independent parameters: curvature magnitude $k_{ij}$, bending-plane orientation $\varphi_{ij}$, and axial twist per unit length $\tau_{ij}$. These parameters define the local bending angle $\theta_{ij} = k_{ij} \cdot \delta_i$ and twist angle $\gamma_{ij} = \tau_{ij} \cdot \delta_i$, where $\delta_i = L_i/n_i$ denotes the uniform arc length of each vertebra in segment $i \in \{1, 2\}$ with total length $L_i$ and $n_i = 7$ vertebrae. The vertebrae are indexed by $j = 1, \dots, n_i$, representing their position along the arc of segment $i$. This fine-grained discretization enhances the geometric fidelity of the model and improves the accuracy of inverse-kinematics prediction and motion generation [25].

A global base frame is fixed at the robot's origin, and local coordinate frames are assigned to each vertebra to track its spatial configuration. The detailed vertebra-level parameterization is depicted in Fig. 2(a), showing the local coordinate frames attached to each subsegment, the bending and twist angles, and the uniform arc length $\delta_i$. The first segment's shape is described by the set of per-vertebra bending angles $\theta_{ij}$, calculated from the local curvature $k_{ij}$ and segment length $\delta_i$. The twist angle $\gamma_{ij}$ is obtained from the twist rate $\tau_{ij}$ over the same arc length, while the bending direction is defined by $\varphi_{ij}$. This local representation enables accurate reconstruction of spatially varying shapes and consistent mapping between tendon actuation and end-effector pose.

Figure 2(b) provides a detailed view of two adjacent vertebrae, illustrating how bending and twisting are jointly represented within the extended PCC formulation. The twist angle $\gamma_{ij}$ is computed from the local twist rate $\tau_{ij}$ over the arc length $\delta_i$, while the bending direction $\theta_{ij}$ defines the orientation of the bending plane for each vertebra. This detailed subsegment depiction clarifies

the geometric relationship between curvature, bending orientation, and torsional deformation, forming the foundation for the vertebra-level kinematic modeling used throughout this work.

**2.1 Tendon Kinematics at the Vertebral Level.** The relationship between the robot's shape and the required tendon displacements is fundamental to achieving accurate actuation. Let (r) denote the radial distance of each tendon from the backbone's central axis. Each segment is actuated by four tendons placed symmetrically around the backbone, with angular positions offset by 45 deg between consecutive segments. The full tendon layout consists of eight tendons positioned $\alpha_k = \{0 \text{ deg}, 45 \text{ deg}, 90 \text{ deg}, 135 \text{ deg}, 180 \text{ deg}, 225 \text{ deg}, 270 \text{ deg}, 315 \text{ deg}\}$ is shown in Figs. 3(a) and 3(b). All tendon projections are computed based on the local bending-plane orientation and the tendon's angular position within the vertebral coordinate frame.

For each vertebra j in segment i, the length change of tendon k, denoted $\Delta l_{ijk}$, is influenced by both bending and twisting. The total change is expressed as:

$$ \Delta l_{ijk} = \beta_{ijk} + \epsilon_{ijk} \quad (1) $$

The bending-induced term $\beta_{ijk}$ is derived from the (PCC) assumption and is defined as:

$$ \delta_i[1 - \kappa_{ij} r \cos(\varphi_{ij} - \alpha_k)] \quad (2) $$

Here, $\delta_i$ denotes the arc length of the subsegment, $\kappa_{ij}$ is the local curvature of the vertebra j, and the cosine projection represents the effective shortening or elongation of the tendon path with respect to the neutral axis (Fig. 3(c)). The second term, $\epsilon_{ijk}$, accounts for the additional deviation due to twist and is defined by:

$$ \epsilon_{ijk} = r \tau_{ij} \sin(\varphi_{ij} - \alpha_k) \quad (3) $$

This term captures the helical path deformation resulting from axial twist $\tau_{ij}$, which affects tendons asymmetrically based on

<figure><img src="image_4.png" /><figcaption>Fig. 1 Two-segment continuum robot, where each segment is modeled as a smooth arc defined by global bending parameter ($\theta_i, \varphi_i$)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2 (a) Fine-grained model with local vertebra frames {$T_{ij}$}, enabling independent shape representation for each vertebra. (b) Zoomed subsegment view showing vertebra-level bending plane $\varphi_{ij}$, arc angle $\theta_{ij}$, twist angle $\gamma_{ij}$, and fixed arc length $\delta_i = L_i/n_i$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 3 (a) First segment of continuum robot with bending plane, tendon layout, and global and local coordinates (b) tendons are symmetrically distributed with 45-deg offsets across vertebra (c) geometric projection of bending angle on tendon locations</figcaption></figure>

their position. The magnitude of this contribution depends on the tendon's angular location relative to the bending plane.

The total length change of tendon k in segment i is obtained by summing the contributions from all $n_i$ vertebrae:

$$ \Delta l_{ik} = \sum_{j=1}^{n_i} \Delta l_{ijk} \\ = \sum_{j=1}^{n_i} \{ \delta_i [1 - \kappa_{ij} r \cos(\varphi_{ij} - \alpha_k)] + r \tau_{ij} \sin(\varphi_{ij} - \alpha_k) \} \quad (4) $$

This summation provides the necessary actuation input for each tendon to achieve the desired configuration of segment i.

## 2.2 Forward Kinematic Model.
Based on the vertebra-level parameterization introduced in Sec. 2, the Forward Kinematic Model (FKM) computes the cumulative end-effector pose by sequentially composing the spatial transformations of all subsegments. For each vertebra, the local transformation is defined using the curvature magnitude $\kappa_{ij}$, bending-plane orientation $\varphi_{ij}$, and axial twist $\tau_{ij}$.

The transformation for each vertebra consists of a rotation $R_z(\varphi_{ij})$ to align the local frame with the bending plane, followed by a constant-curvature bending rotation $R_y(\theta_{ij})$, and a final twist rotation $R_z(\tau_{ij} - \varphi_{ij})$ to apply the axial torsion relative to the bending direction.

The position vector of each subsegment is defined as follows:

$$ P_{ij} = \frac{1}{k_{ij}} \begin{bmatrix} (1 - \cos \theta_{ij}) \cos \varphi_{ij} \\ (1 - \cos \theta_{ij}) \sin \varphi_{ij} \\ \sin \theta_{ij} \end{bmatrix} \quad (5) $$

The homogeneous transformation matrix $T_{ij} \in SE(3)$ for vertebra j in segment i is constructed as:

$$ T_{ij} = \begin{bmatrix} R_z(\varphi_{ij}) & R_y(\theta_{ij}) & R_z(\tau_{ij} - \varphi_{ij}) & P_{ij} \\ 0_{1 \times 3} & 1 & 0 & 0 \end{bmatrix} \quad (6) $$

The global pose of the robot tip is obtained by chaining all transformations from the base to the end effector:

$$ T_{end} = \prod_{j=1}^{n_1} T_{1j} \cdot \prod_{j=1}^{n_2} T_{2j} = \prod_{i=1}^{2} \prod_{j=1}^{7} T_{ij} \quad (7) $$

This formulation captures both bending and torsional deformations and enables precise shape reconstruction and task-space mapping of the TDCR.

# 3 Inverse-Kinematics Framework: Learning, Hybrid Refinement, and Actuation

The inverse-kinematics model (IKM) for (TDCRs) involves determining the internal configuration parameters $q = \{\kappa_{ij}, \varphi_{ij}, \tau_{ij}\}$ for each vertebra j in segment i that produce a desired end-effector pose $T_{des} \in SE(3)$. This problem is commonly posed as a constrained nonlinear least-squares optimization:

$$ \min_q \|P_{end} - P_{des}\|^2 + \omega \|R_{end} - R_{des}\|_F^2 \quad (8) $$

Subject to:

$$ 0 \le \kappa_{ij} \le \kappa_{max}, |\varphi_{ij}| \le \varphi_{max}, |\tau_{ij}| \le \tau_{max} $$

Here, $P_{des} \in \mathbb{R}^3$ and $R(des) \in SE(3)$ the position and rotation matrix components of the forward kinematics $T_q$ and $\omega \in [0, 1]$ balances position versus orientation accuracy.

In this work, the orientation discrepancy is expressed through the Frobenius norm of the rotation–matrix difference $\|R_{end} - R_{des}\|_F^2$, which provides a compact Euclidean approximation of rotational deviation in $SO(3)$. This chordal distance is monotonic with the geodesic angle on $SO(3)$ leading to the same minimizer while offering smoother gradients and lower computational cost. Although this measure does not explicitly compute the geodesic angle, it offers sufficient sensitivity for orientation matching in quasi-static inverse-kinematics problems and is computationally efficient for large-scale optimization. A weighting value of $\omega = 0.05$ was empirically selected to achieve a balanced tradeoff, ensuring that the optimization primarily minimizes positional error while still maintaining acceptable orientation precision (sub-degree range). Larger weights ($\omega > 0.1$) were found to slow convergence without notable improvement in accuracy, whereas smaller weights ($\omega < 0.02$) reduced orientation consistency. We also empirically probed $\omega$ in a narrow neighborhood around 0.05 and observed consistent behavior across all 16 trajectory, constraint scenarios, confirming $\omega = 0.05$ as a robust choice for our objective scaling. This formulation ensures physical feasibility by enforcing geometric constraints on curvature, bending angle, and twist, while maintaining numerical stability and moderate computational cost. However, its use in fully real-time applications remains limited due to the inherent complexity of the nonlinear optimization [37].

In this study, the optimization problem is formulated primarily to solve the inverse kinematics (IK) of the tendon-driven continuum robot, where the tendon-length variations are derived from the joint-level configuration parameters curvature $\kappa_{ij}$, bending-plane angle $\varphi_{ij}$, and twist rate $\tau_{ij}$ for each vertebra along the backbone. The optimization bounds were defined to ensure physically
<figure><img src="image_7.png" /><figcaption>Fig. 4 Neural network architecture for IK: mapping end-effector pose to a 42-dimensional configuration vector</figcaption></figure>

<table><caption>Table 2 Training performance of MLP and DNN on the circular trajectory with fixed tilt-Y (N = 1000 validation samples; 200 epochs; Adam)</caption><thead><tr><th>Metric</th><th>MLP</th><th>DNN</th></tr></thead><tbody><tr><td>Final validation RMSE</td><td>0.7997</td><td>0.6311</td></tr><tr><td>Training epochs</td><td>200</td><td>200</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.001</td></tr><tr><td>Batch size</td><td>32</td><td>32</td></tr><tr><td>Convergence behavior</td><td>Stable</td><td>Smoother and faster</td></tr><tr><td>Validation variance</td><td>Moderate</td><td>Low</td></tr><tr><td>Generalization quality</td><td>Acceptable</td><td>High</td></tr></tbody></table>

Note: The threshold of ~7 Hz corresponds to the typical range (5–10 Hz) reported in prior real-time control studies for tendon-driven continuum robots [20,30].

<table><caption>Table 3 Benchmark results of neural and hybrid IK models under the lock X-axis constraint (N = 1000 samples, 1 epoch training)</caption><thead><tr><th>Model</th><th>Train Time (s, 1 epoch)</th><th>Inference Time (ms, sample)</th><th>Val RMSE</th><th>Num Params</th></tr></thead><tbody><tr><td>MLP</td><td>0.372</td><td>0.17</td><td>9.89</td><td>7722</td></tr><tr><td>DNN</td><td>0.440</td><td>0.23</td><td>10.42</td><td>29,162</td></tr><tr><td>HybMLP</td><td>—</td><td>21.10</td><td>6.52</td><td>7722</td></tr><tr><td>HybDNN</td><td>—</td><td>17.86</td><td>5.85</td><td>29,162</td></tr></tbody></table>

meaningful configurations and stable numerical convergence. For all vertebrae, the parameter limits were set as:

$$ \kappa_{ij} \in [0, 20.0] \text{ m}^{-1}, \quad \phi_{ij} \in [0, 2\pi] \text{rad}, \quad \tau_{ij} \in [-5.0, 5.0] \text{rad.m}^{-1} $$

These bounds define the feasible joint-space range for curvature, bending, and twist variables across all 14 vertebrae and prevent nonphysical deformations during the optimization process. Additionally, actuator travel is restricted by tendon-length changes computed from Eq. (4), i.e., $|\Delta l_k| \le 40$ mm for all tendons k, consistent with the hardware limits specified in Sec. 5.3. Collision avoidance was not enforced in this IK study; trajectories were confined to a nominal workspace envelope aligned with the robot's kinematic reach. All workspace tracking tasks were therefore performed within this nominal kinematic workspace, without obstacles or external collision constraints, as the primary objective was to assess the hybrid IK solver's accuracy and stability under bounded joint-space variations.

**3.1 Supervised Learning for Rapid Inverse-Kinematics Model Approximation.** To overcome runtime limitations, a supervised learning approach is employed to approximate the

IKM mapping. A feedforward neural network receives the end-effector pose $T_{des}$, represented by a 3D position vector and a flattened nine-element rotation matrix. The output is a 42-dimensional configuration vector, representing all vertebral parameters for a 14-vertebra TDCR. The neural network architectures were implemented (Fig. 4):

* MLP: Three-layer, 64 neurons per layer and ReLU activation;

* DNN: Four layers (128-128-64-64 neurons), using ReLU activation.

Training used a dataset of N = 16,000 pose-configuration pairs generated via forward kinematics and constrained optimization (fmincon). The model was trained using the Adam optimizer (learning rate = 0.001, batch size = 32) for 200 epochs, minimizing mean squared error:

$$ \sigma = \frac{1}{N} \sum_{i=1}^{N} \| q_{i}^{\text{pred}} - q_{i}^{\text{true}} \|^{2} \quad (9) $$

where each output $q_i \in \mathbb{R}^{42}$ represents the full vertebral parameter set $q = \{\kappa_{ij}, \phi_{ij}, \tau_{ij}\}$ for all vertebrae in the TDCR. Although neural networks provide rapid inference and generalization, they do not
<figure><img src="image_8.png" /><figcaption>Fig. 5 Circular trajectory tracking with the hybrid IK framework. (a) HybMLP versus target path; (b) HybDNN versus target path; (c) 3D TDCR configuration at sample 617/1000 along the path. Dashed black = target path; rainbow = HybDNN path (color encodes progress).</figcaption></figure>

inherently guarantee constraint satisfaction, particularly near work-space boundaries or under strict orientation requirements.

The neural models were trained using the Adam optimizer [48] with default parameters ($\beta_1 = 0.9$, $\beta_2 = 0.999$, learning rate = $10^{-3}$) as implemented in MATLAB R2023b. The subsequent constrained refinement employed MATLAB's *fmincon* solver (interior-point algorithm) to ensure curvature and orientation feasibility.

Two network architectures were considered to evaluate the influence of depth and parameter count: a three-layer MLP (64 neurons per layer, ReLU activation) and a four-layer DNN (128–128–64–64 neurons, ReLU activation). The DNN architecture was selected as the main estimator due to its higher representation capacity and smoother convergence behavior, while maintaining reasonable computational cost. This design choice balances generalization

<figure><img src="image_9.png" /><figcaption>Fig. 6 Elliptical trajectory tracking with the hybrid IK framework. (a) HybMLP versus target path; (b) HybDNN versus target path; (c) 3D TDCR configuration at sample 463/1000 along the path. Dashed black = target path; rainbow = HybDNN path (color encodes progress).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7 Helical-trajectory tracking with the hybrid IK framework. (a) HybMLP versus target path; (b) HybDNN versus target path; (c) 3D TDCR configuration at sample 969/1000 along the path. Dashed black = target path; rainbow = HybDNN path (color encodes progress).</figcaption></figure>

accuracy and training efficiency, providing a solid initialization for the hybrid optimization stage.

**3.2 Hybrid Inverse-Kinematics Model and Tendon Actuation Mapping.** To address the limitations of pure learning, a hybrid IKM approach is introduced that couples fast inference with constraint-enforced optimization. The method proceeds in two stages:

(1) *Neural Inference*: A trained MLP or DNN provides an initial $q^{NN} = \{k_{ij}^{NN}, \phi_{ij}^{NN}, \tau_{ij}^{NN}\}$ from $T_{des}$.

(2) *Optimization Refinement*: A constrained nonlinear least-squares problem is solved as formulated in Eq. (8).

This hybrid approach ensures constraint satisfaction while benefiting from fast convergence. It is particularly effective in high-curvature and tight orientation configurations. Once the optimized configuration $q^* = \{k_{ij}^*, \phi_{ij}^*, \tau_{ij}^*\}$ obtained from hybrid approach, tendon displacements $\Delta I_{ik}^*$ are computed by Eq. (4). These displacements provide the final motor commands for actuation.

## 4 Experimental Setup

To evaluate and train the proposed hybrid (IK) framework, simulations were performed using the two-segment tendon-driven continuum robot (TDCR) described in Sec. 2, consisting of a total of 14 vertebrae modeled under the PCC formulation with axial twist extension.

Four representative task-space trajectories, circular-, elliptical-, helical-, and butterfly-shaped, were designed to span diverse motion patterns, and each trajectory was combined with four distinct end-effector orientation constraints: Lock X-axis, Lock Y-axis, Lock Z-axis (upright), and Fixed Tilt around Y, resulting in sixteen trajectory constraint configurations. For each configuration, a dataset of 1000 samples was generated by solving the constrained IK optimization problem formulated in Sec. 3.2, within the physically admissible bounds of the vertebra parameters: $\kappa_{ij} \in [0, 20.0]$, $\phi_{ij} \in [0, 2\pi]$, and $\tau_{ij} \in [-5.0, 5.0]$. The cost

function used a weighting factor of $\omega = 0.05$ to balance position and orientation accuracy.

This procedure ensured that all generated samples corresponded to feasible backbone configurations, satisfying geometric constraints on curvature, bending, and twist. In total, 16,000 IK solutions were obtained through forward kinematics followed by constrained optimization. The resulting dataset provides physically consistent mappings between end-effector poses and tendon-length actuation variables, ensuring robust generalization of the neural networks (MLP, DNN, and hybrid models). All experiments were implemented in MATLAB R2023b, using the Deep Learning Toolbox for network training and fmincon for the hybrid refinement.

These specifications allow reproducibility of the dataset generation and provide a standardized basis for benchmarking hybrid learning-based IK frameworks.

## 5 Results

To evaluate training performance, both the MLP and DNN architectures were monitored using a representative validation case: the circular trajectory under the “Fixed Tilt around Y” constraint. As reported in Table 2, the DNN achieved a lower final validation RMSE (0.6311 versus 0.7997), exhibited smoother convergence, and demonstrated reduced loss variance. These results indicate improved generalization, motivating its selection as the default estimator in the hybrid pipeline.

### 5.1 Hybrid Inverse-Kinematics Performance.
The hybrid IK achieved higher accuracy than its purely neural counterparts, at the cost of a modest computational overhead. The baseline MLP and DNN models (one-epoch warm-up) required 0.372 s and 0.440 s of training time, respectively, and achieved validation RMSE values of 9.89 and 10.42. Their per-sample inference times were extremely low (0.17 ms for MLP and 0.23 ms for DNN), reflecting the efficiency of purely feedforward estimation.

By contrast, the hybrid configurations (HybMLP and HybDNN) incorporate a lightweight constrained-optimization refinement
<figure><img src="image_11.png" /><figcaption>Fig. 8 Butterfly trajectory tracking with the hybrid IK framework. (a) HybMLP versus target path; (b) HybDNN versus target path; (c) 3D TDCR configuration at sample 548/1000 along the path. Dashed black = target path; rainbow = HybDNN path (color encodes progress).</figcaption></figure>

<table><caption>Table 4 Absolute position and orientation tracking errors (sample case: circular trajectory, lock Z-axis constraint)</caption><thead><tr><th>Model</th><th>Mean Pos. Error (mm)</th><th>Max Pos. Error (mm)</th><th>Mean Orient. Error (deg)</th><th>Max Orient. Error (deg)</th></tr></thead><tbody><tr><td>MLP</td><td>79.296</td><td>278.456</td><td>2.0707</td><td>2.8284</td></tr><tr><td>DNN</td><td>86.965</td><td>282.566</td><td>1.6115</td><td>2.8283</td></tr><tr><td>HybDNN (this work)</td><td>0.000293</td><td>0.00439</td><td>0.000007</td><td>0.000132</td></tr></tbody></table>

applied after the neural prediction. Although this stage does not involve additional training (hence the omitted training-time entries), it introduces a small runtime cost, increasing inference time to ≈21.10 ms for HybMLP and ≈17.86 ms for HybDNN. In return, both hybrid models achieve substantially lower validation errors, 6.52 for HybMLP and 5.85 for HybDNN, corresponding to accuracy gains of approximately 34% and 44%, respectively, relative to their neural baselines. The computational efficiency and prediction accuracy of the baseline and hybrid IK models are summarized in Table 3.

These results confirm that the hybrid refinement consistently enhances prediction fidelity and constraint satisfaction while maintaining practical computational feasibility for near-real-time TDCR motion generation. Figures 17–20 illustrate this trend for the circular trajectory under the Lock X-axis constraint, where the hybrid variants exhibit smaller mean position and orientation errors with lower variance across the 1000-sample test set. The improvement remains consistent across other trajectory and constraint conditions (N = 16,000 overall).

The optimization stage becomes particularly beneficial in high-curvature regions ($k \ge 0.018 \text{ m}^{-1}$) or under tight orientation constraints (>40 deg), where purely learned models may violate curvature or tendon-length limits. In such cases, the hybrid refinement effectively corrects infeasible predictions, as exemplified by the helical-trajectory results (Fig. 7(c) and Figs. 25–28).

All computations were executed sequentially in MATLAB R2023b on a standard workstation (Intel Core i5, 4 cores, 9.5 GB RAM, Windows 10) without GPU acceleration, ensuring consistent timing comparisons.

## 5.2 Task-Space Trajectory Tracking Accuracy.
The proposed hybrid (IK) framework demonstrated consistent and accurate task-space tracking across all evaluated trajectories and orientation constraints.

Figures 5–8 summarize the representative results for the four trajectory types: circular, elliptical, helical, and butterfly under the Fixed Tilt around Y constraint. In all cases, the predicted end-effector paths closely matched the target trajectories, confirming the feasibility and precision of the hybrid approach. The DNN-based hybrid solution exhibited smoother motion profiles and smaller residual deviations compared with the MLP-based counterpart, particularly in regions of high-curvature or rapid orientation change.

Quantitatively, the hybrid models reduced position and orientation errors by several orders of magnitude relative to standalone neural estimators.

For example, in the circular trajectory under the Lock Z-axis constraint (Table 4), the mean position error drops from 79.296 mm (MLP) to 0.000293 mm (HybDNN) and the mean orientation error from 2.0707 deg to 0.000007 deg. Across all four orientation constraints (Table 5), baseline errors in the tens of millimeters/degrees range are consistently brought to the sub-millimeter (often micro-level)/sub-degree (often micro-degree) range.

## 5.3 Tendon Actuation Profiles.
Figures 9–16 depict the tendon actuation profiles for both robot segments under the “Fixed Tilt around Y” constraint. For all tested trajectories, tendon-
<table><caption>Table 5 Comparison of mean position and orientation tracking errors (units: mm, deg) between baseline neural models (MLP and DNN) and their hybrid counterparts for the circular trajectory under four end-effector orientation constraints</caption><thead><tr><th>Orientation Constraint</th><th>Mean Pos. Err—MLP</th><th>Mean Pos. Err—Hyb MLP</th><th>Mean Ori. Err—MLP</th><th>Mean Ori. Err—Hybrid MLP</th><th>Mean Pos. Err—DNN</th><th>Mean Pos. Err—Hyb DNN</th><th>Mean Ori. Err—DNN</th><th>Mean Ori. Err—Hybrid DNN</th></tr></thead><tbody><tr><td>Lock X-axis</td><td>69.823</td><td>0.001592</td><td>1.4613</td><td>0.000212</td><td>69.165</td><td>0.001362</td><td>1.3474</td><td>0.00014</td></tr><tr><td>Lock Y-axis</td><td>85.587</td><td>0.004867</td><td>1.8630</td><td>0.000315</td><td>89.368</td><td>0.005412</td><td>1.9245</td><td>0.00034</td></tr><tr><td>Lock Z-axis (upright)</td><td>79.296</td><td>0.00649</td><td>2.0707</td><td>0.000008</td><td>86.965</td><td>0.000293</td><td>1.6115</td><td>0.00001</td></tr><tr><td>Fixed Tilt around Y</td><td>68.599</td><td>0.000251</td><td>1.2398</td><td>0.000006</td><td>63.001</td><td>0.000258</td><td>1.2351</td><td>0.00001</td></tr></tbody></table>

Note: All improvements are statistically significant ($p \ll 0.01$).

length variations remained within hardware limits (±4 cm) and displayed smooth, coordinated patterns. Segment 2 tendons exhibited larger excursions, especially for helical and butterfly paths, highlighting their role in handling distal complexity. The actuation profiles remained physically feasible under all orientation constraints (Lock X, Y, Z, and Fixed Tilt around Y), confirming the generalizability of the hybrid IK approach. Even under high-curvature paths, the framework ensured actuation-level realizability across all segments and conditions.

**5.4 Comparative Tracking Accuracy.** Figures 17–32 show position and orientation tracking errors across four trajectories and four orientation constraints. The hybrid models (HybMLP and HybDNN) significantly outperformed their standalone counterparts in both mean and variance of errors.

*   **Circular/Elliptical:** Hybrid models notably improved orientation precision, especially under Lock Y and Tilt-Y constraints. HybDNN showed the most consistent results.

*   **Helical/Butterfly:** These complex paths posed greater challenges, but the hybrid models maintained sub-millimeter position and sub-degree orientation errors, with HybDNN again yielding the best results.

To provide a concise quantitative evaluation, the absolute position and orientation tracking errors were computed in physical units (millimeters and degrees). Considering the variety of trajectory-constraint combinations (four trajectory types: circular, elliptical, helical, and butterfly) and four end-effector orientation modes (Fixed Tilt around Y, Lock X-axis, Lock Y-axis, and Lock Z-axis (upright)), a total of 16 test scenarios were analyzed.

The overall results demonstrate that the proposed hybrid inverse-kinematics framework achieves consistent and statistically

significant accuracy improvements compared with standalone neural estimators. Across all evaluated trajectories and orientation constraints, the hybrid models exhibited mean position errors below 0.01 mm and mean orientation errors below 0.001 deg, representing improvements exceeding three orders of magnitude relative to the baseline MLP and DNN networks. The pairwise statistical tests summarized in Table 4 confirmed that these improvements are highly significant ($p \ll 0.01$) for both position and orientation errors across all four end-effector orientation constraints of the circular trajectory.

Similar trends were observed for the elliptical, helical, and butterfly trajectories, indicating that the hybrid learning-optimization architecture generalizes robustly to different motion patterns and constraint conditions. These findings validate the effectiveness of combining neural prediction with constraint-aware optimization, providing both the high inference speed of learning-based methods and the physical consistency of optimization-based solvers, making the framework suitable for real-time, high-precision control of tendon-driven continuum robots.

Detailed $p$-values corresponding to the two-sample $t$-tests between baseline and hybrid models are provided in Table 6 (Appendix).

**5.5 Tendon Displacement Across Trajectories and Orientation Constraints.** Figures 33–36 provide an overview of the mean and variance of tendon displacements across both segments for all trajectories and orientation constraints. Segment 2, particularly tendons T1 and T3, exhibited higher displacement variance correlated with high-curvature regions and tight orientation constraints. The “Fixed Tilt around Y” constraint generally led to smoother tendon transitions and lower actuation effort.

<figure><img src="image_12.png" /><figcaption>Fig. 9 Tendon-length variations in Segment 1 during circular trajectory tracking</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10 Tendon-length variations in Segment 2 during circular trajectory tracking</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Fig. 11 Tendon-length variations in Segment 1 during elliptical trajectory tracking</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14 Tendon-length variations in Segment 2 during helical-trajectory tracking</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 12 Tendon-length variations in Segment 2 during elliptical trajectory tracking</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15 Tendon-length variations in Segment 1 during butterfly-shaped trajectory tracking</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 13 Tendon-length variations in Segment 1 during helical-trajectory tracking</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 16 Tendon-length variations in Segment 2 during butterfly-shaped trajectory tracking</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. 17 Mean and variance of position tracking errors for circular trajectory execution using MLP and deep neural networks under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 20 Orientation error mean and variance during circular trajectory tracking using hybrid models (HybMLP and HybDeep) under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 18 Orientation error mean and variance during circular trajectory tracking using MLP and deep neural networks under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 21 Mean and variance of position tracking errors during elliptical trajectory tracking using MLP and Deep networks under four orientation constraints</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 19 Position error mean and variance during circular trajectory tracking using hybrid models (HybMLP and HybDeep) under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 22 Orientation error mean and variance during elliptical path tracking using MLP and Deep models across four orientation constraints</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Fig. 23 Mean and variance of position tracking errors during elliptical trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 26 Mean and variance of orientation tracking errors during helical-trajectory tracking using MLP and deep neural networks under four orientation constraints</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 24 Mean and variance of orientation tracking errors during elliptical trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 27 Mean and variance of position tracking errors during helical-trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Fig. 25 Mean and variance of position tracking errors during helical-trajectory execution using MLP and deep neural networks under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 28 Mean and variance of orientation tracking errors during helical-trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /><figcaption>Fig. 29 Mean and variance of position tracking errors during butterfly trajectory tracking using MLP and deep neural networks under four orientation constraints</figcaption></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /><figcaption>Fig. 32 Mean and variance of orientation tracking errors during butterfly trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>

<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /><figcaption>Fig. 30 Mean and variance of orientation tracking errors during butterfly trajectory tracking using MLP and deep neural networks under four orientation constraints</figcaption></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /><figcaption>Fig. 33 Mean and variance of tendon length changes in both segments during circular trajectory tracking under four end-effector orientation constraints</figcaption></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /><figcaption>Fig. 31 Mean and variance of position tracking errors during butterfly trajectory tracking using hybrid models (HybMLP and HybDeep) under four orientation constraints</figcaption></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /><figcaption>Fig. 34 Mean and variance of tendon length changes across all tendons and segments during elliptical trajectory tracking under fixed end-effector orientation constraints</figcaption></figure>
<figure><img src="image_44.png" /><figcaption>Fig. 35 Mean and variance of tendon length changes across all tendons and segments during helical-trajectory tracking under fixed end-effector orientation constraints</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>Fig. 36 Mean and variance of tendon length changes across all tendons and segments during butterfly trajectory tracking under fixed end-effector orientation constraints</figcaption></figure>

These findings confirm that the proposed hybrid (IK) strategy generates actuation profiles that are not only accurate but also mechanically consistent and suitable for implementation on physical platforms.

## 6 Conclusion

This paper presented a hybrid inverse-kinematics framework for tendon-driven continuum robots that couples fast supervised learning with constraint-aware optimization to deliver feasible and accurate trajectory tracking under challenging spatial and orientation constraints. Training data were derived from optimal IK solutions over sixteen trajectory-constraint combinations (four trajectories × four orientation modes), ensuring physics-consistent supervision. Across extensive simulations on a two-segment TDCR, the hybrid approach achieved statistically significant gains ($p \le 0.01$) and maintained sub-millimeter position and sub-degree orientation accuracy with stable actuation profiles, indicating suitability for near-real-time surgical and industrial tasks. The current study adopts a quasi-static, PCC-based formulation focused on geometric and actuation constraints. Looking ahead, we will incorporate a

dynamic Cosserat-rod formulation with closed-loop control (e.g., disturbance-aware feedback and constraint-consistent refinement), and extend constraint handling to dynamic obstacle avoidance and smoothness optimization. We will also pursue robust sim-to-real transfer via adaptive learning on multi-segment platforms.

## Funding Data

* Mohammad Jabari has been fully supported by the PNRR-NGEU project, which has received funding from the MUR—DM 351/2022.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent was obtained for all individuals. Documentation provided upon request. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Appendix—Statistical Significance Analysis

All results indicate statistically significant improvements of the hybrid models over their baseline neural counterparts for both position and orientation accuracy ($p \ll 0.01$).

<table><caption>Table 6 p-Values from two-sample t-tests between baseline and hybrid models (circular trajectory)</caption><thead><tr><th>Orientation constraint</th><th>p (Position, MLP)</th><th>p (Orientation, MLP)</th><th>p (Position, DNN)</th><th>p (Orientation, DNN)</th></tr></thead><tbody><tr><td>Lock X-axis</td><td>2.9 × 10<sup>-248</sup></td><td>1.2 × 10<sup>-294</sup></td><td>2.0 × 10<sup>-210</sup></td><td>5.6 × 10<sup>-284</sup></td></tr><tr><td>Lock Y-axis</td><td>3.7 × 10<sup>-267</sup></td><td>1.1 × 10<sup>-289</sup></td><td>4.4 × 10<sup>-243</sup></td><td>2.0 × 10<sup>-288</sup></td></tr><tr><td>Lock Z-axis (upright)</td><td>8.2 × 10<sup>-235</sup></td><td>5.9 × 10<sup>-298</sup></td><td>6.8 × 10<sup>-251</sup></td><td>3.1 × 10<sup>-295</sup></td></tr><tr><td>Fixed tilt around Y</td><td>1.4 × 10<sup>-254</sup></td><td>7.6 × 10<sup>-292</sup></td><td>3.2 × 10<sup>-267</sup></td><td>1.9 × 10<sup>-290</sup></td></tr></tbody></table>

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