In the textile industry, the winding process is a critical step that bridges spinning and weaving, where bobbins (or yarn tubes) must be efficiently transferred to winders. Traditional manual handling is labor-intensive and prone to inefficiencies, while existing automated end effectors often suffer from unstable gripping and poor adaptability to multi-size bobbins. To address these challenges, we propose a novel underactuated bionic end effector inspired by human hand mechanics. This end effector features three fingers with tendon-driven mechanisms and passive compliance, enabling stable grasping of bobbins with diameters ranging from 18 mm to 37 mm, as well as larger objects. In this work, we detail the design, analysis, and experimental validation of this end effector, focusing on its ability to achieve both fingertip and enveloping grasps adaptively.
Our design philosophy centers on mimicking the human hand’s functional mechanisms, particularly the tendon-pulley system that allows for dexterous and adaptive movements. The end effector is intended for use in automated bobbin loading systems, where it must handle various bobbin sizes without damaging the yarn. We start by outlining the structural design of the end effector, followed by a thorough analysis of gripping forces, spring optimization for passive compliance, and stability assessment using contact mechanics. Finally, we present experimental results from prototype testing, demonstrating high success rates in grasping tasks. Throughout this article, we emphasize the role of the end effector in enhancing textile automation, and we incorporate multiple tables and equations to summarize key findings.
The human hand’s grasping capability stems from its complex musculoskeletal system, where tendons transmit forces from muscles to finger bones via pulleys (sheaths). This biological principle informs our end effector design: we use tendon-rope systems coupled with pulleys to drive finger joints, while springs provide restoring forces for extension. This configuration enables underactuation, meaning fewer actuators than degrees of freedom, which simplifies control and improves adaptability. The end effector consists of three fingers arranged in a triangular pattern on a palm, each finger comprising proximal, medial, and distal phalanges connected by rotational joints. A single tendon, driven by a stepper motor, flexes all joints simultaneously, whereas linear springs at each joint facilitate extension and passive compliance. This design allows the end effector to switch between fingertip and enveloping grasps based on object contact, without active control intervention.
To quantify the dimensions, we base finger segment lengths on human hand proportions, tailored for bobbin grasping. The parameters are summarized in Table 1. Each finger module is 44 mm in the proximal phalanx, 35 mm in the medial phalanx, and 30 mm in the distal phalanx, with a uniform width of 23 mm and thickness of 26 mm. The entire end effector has a height of 210 mm, width of 165 mm, and length of 165 mm, making it compact for industrial settings. The triangular finger distribution ensures force closure and stability during grasping, a key advantage of this end effector over two-finger designs.
| Segment | Length (mm) | Width (mm) | Thickness (mm) | Mass (g) |
|---|---|---|---|---|
| Proximal Phalanx | 44 | 23 | 26 | 169 |
| Medial Phalanx | 35 | 23 | 26 | 107 |
| Distal Phalanx | 30 | 23 | 26 | 100 |
The tendon routing in our end effector is critical for force transmission. As shown in the design schematic, the tendon passes through pulleys at each joint, with larger pulleys at joints to change force direction and smaller pulleys at turning points. The tendon tension $F_a$ generates joint torques, while spring forces oppose flexion. For joint $i$ (where $i=1,2,3$ for proximal, medial, and distal joints, respectively), the torque $\tau_i$ is given by:
$$\tau_i = – (F_a R_i – K_i w_i^2 \Delta \eta_i)$$
Here, $R_i$ is the pulley radius at joint $i$, $K_i$ is the spring stiffness, $w_i$ is the moment arm of the spring force, and $\Delta \eta_i$ is the angular displacement. The negative sign indicates that flexion torque opposes spring extension. This equation forms the basis for our grip force analysis. The end effector’s ability to exert sufficient force on bobbins is essential for stable grasping, and we derive the fingertip contact force $F_3$ for the distal phalanx as:
$$F_3 = \frac{F_a R_3 – K_3 w_3^2 \Delta \eta_3}{M_3}$$
where $M_3$ is the moment arm for the torque at the distal joint. This shows that contact force decreases with higher spring stiffness, highlighting the need for careful spring selection in the end effector design.
Spring configuration plays a pivotal role in achieving passive compliance and proper finger movement sequencing in our end effector. Without optimal springs, fingers may curl prematurely, leading to inefficient grasps. We enforce the condition $|\tau_1| \geq |\tau_2| \geq |\tau_3|$ to ensure that the proximal joint moves first, followed by medial and distal joints, mimicking human grasping. From the torque equation, this implies $K_1 \leq K_2 \leq K_3$. We select linear springs from standard options, considering constraints such as maximum load ($f_{\text{max}} = 8\,\text{N}$), working load ($f = 5\,\text{N}$), and stroke ($h = 6\,\text{mm}$). Using 65 Mn spring steel with wire diameter $d = 0.8\,\text{mm}$, mean coil diameter $D = 9\,\text{mm}$, and number of active coils $n = 11$, the stiffness $K$ is calculated as:
$$K = \frac{G d^4}{8 n D^3}$$
where $G = 79,000\,\text{MPa}$ is the shear modulus. This yields $K \approx 0.504\,\text{N/mm}$. We test four stiffness configurations (A to D) to optimize the grasping workspace, as summarized in Table 2. Configuration D, with $K_1 = 0.35\,\text{N/mm}$, $K_2 = 0.40\,\text{N/mm}$, and $K_3 = 0.50\,\text{N/mm}$, provides the largest workspace and is chosen for our end effector. This configuration enables adaptive grasping: if the proximal phalanx contacts an object, the medial and distal phalanges continue to flex for an enveloping grasp; otherwise, the finger performs a fingertip grasp. This passive adaptability is a key feature of our end effector.
| Configuration | $K_1$ (N/mm) | $K_2$ (N/mm) | $K_3$ (N/mm) | Grasping Workspace |
|---|---|---|---|---|
| A | 0.45 | 0.45 | 0.50 | Limited |
| B | 0.45 | 0.50 | 0.50 | Moderate |
| C | 0.50 | 0.50 | 0.50 | Moderate |
| D | 0.35 | 0.40 | 0.50 | Largest |
Stability analysis is crucial for ensuring reliable bobbin grasping with our end effector. We extend existing two-dimensional models to three-dimensional space, considering three fingers in frictional point contact. Using a virtual spring model at each fingertip, we analyze the system’s potential energy $U$ to determine stability conditions. Let $F_i$ be the force from finger $i$ ($i=1,2,3$) at contact point $i$, with all forces intersecting at an internal center point $O$. Define position vectors $\mathbf{r}_i$ from $O$ to contact points, and angles $\theta_i$ between $\mathbf{r}_i$ and the x-axis. Virtual springs with stiffnesses $k_{xi}, k_{yi}, k_{zi}$ are assumed along orthogonal directions at each contact. For infinitesimal displacements $\boldsymbol{\epsilon} = [\epsilon_x, \epsilon_y, \epsilon_z, \xi, \eta, \zeta]^T$ of the object, the spring compressions $\epsilon_{xi}, \epsilon_{yi}, \epsilon_{zi}$ are derived from geometric relations:
$$\epsilon_{xi} = x c_i + y s_i – \|\mathbf{r}_i\| s_i^2 (1 – \cos \xi) – \|\mathbf{r}_i\| c_i^2 (1 – \cos \eta) – \|\mathbf{r}_i\| (1 – \cos \zeta)$$
$$\epsilon_{yi} = -x s_i + y c_i – \|\mathbf{r}_i\| s_i c_i (1 – \cos \xi) + \|\mathbf{r}_i\| s_i c_i (1 – \cos \eta) + \|\mathbf{r}_i\| \sin \zeta$$
$$\epsilon_{zi} = z + \|\mathbf{r}_i\| s_i \sin \xi – \|\mathbf{r}_i\| c_i \sin \eta$$
where $s_i = \sin \theta_i$ and $c_i = \cos \theta_i$. The potential energy is:
$$U = \frac{1}{2} \sum_{i=1}^3 \boldsymbol{\epsilon}_i^T \mathbf{k}_i \boldsymbol{\epsilon}_i + \sum_{i=1}^3 \epsilon_{xi} \|F_i\|$$
with $\boldsymbol{\epsilon}_i = [\epsilon_{xi}, \epsilon_{yi}, \epsilon_{zi}]^T$ and $\mathbf{k}_i = \text{diag}(k_{xi}, k_{yi}, k_{zi})$. Stability requires $U$ to be at a local minimum, which translates to positive definiteness of the Hessian matrix $\mathbf{H}(0)$. After algebraic manipulation, we obtain conditions on normalized stiffness parameters. Let $k_{cx} = \sum k_{xi}$, $k_{cy} = \sum k_{yi}$, $k_{cz} = \sum k_{zi}$, $F_c = \sum \|F_i\|$, and $r_c = \sum \|\mathbf{r}_i\|$, with normalized values $kn_{xi} = k_{xi}/k_{cx}$, $Fn_i = \|F_i\|/F_c$, and $rn_i = \|\mathbf{r}_i\|/r_c$. The stability conditions are:
$$k_{cx} > 0$$
$$k_{cy} > \frac{F_c}{r_c} \left( \sum_{i=1}^3 rn_i Fn_i \right) / \left( \sum_{i=1}^3 rn_i^2 kn_{yi} \right)$$
$$k_{cz} > k_{cz2}$$
where $k_{cz2}$ is derived from:
$$k_{cz2} = \frac{F_c}{r_c} \frac{B + \sqrt{B^2 – 4A b_1 b_2}}{2A}$$
with $A = a_1 a_2 – c^2$, $B = a_1 b_2 + a_2 b_1$, $a_1 = \sum rn_i^2 kn_{zi} s_i^2$, $a_2 = \sum rn_i^2 kn_{zi} c_i^2$, $b_1 = \sum rn_i Fn_i s_i^2$, $b_2 = \sum rn_i Fn_i c_i^2$, and $c = \sum rn_i^2 kn_{zi} s_i c_i$. These inequalities indicate that higher finger stiffness and smaller object radius enhance stability. Moreover, for three-finger grasping, force closure is achieved when the internal force center coincides with the centroid of the contact triangle. Our end effector’s triangular finger arrangement promotes this, improving grip stability. Additionally, fingertip curvature affects stability: larger curvature (smaller radius) increases stiffness eigenvalues, making the end effector more robust to disturbances.
Based on the stability analysis, we optimize the fingertip design for our end effector. Human fingertips combine elliptical and rectangular profiles; we adopt an elliptical curve for smooth transition and high curvature. Using the distal phalanx length as the distance from the joint axis to the fingertip, we set the ellipse semi-major axis $a = 17.82\,\text{mm}$ and semi-minor axis $b = 13\,\text{mm}$, with foci distances satisfying $a + c = 30\,\text{mm}$ and $c^2 = a^2 – b^2$. The ellipse equation is:
$$\frac{x^2}{17.82^2} + \frac{y^2}{13^2} = 1$$
This shape is lofted to create a 3D fingertip model, covered with soft rubber (friction coefficient $\mu = 0.6$) to enhance grip and stability. The high curvature at the contact point aligns with stability requirements, ensuring that the end effector maintains secure holds on bobbins.
To validate our end effector design, we fabricated a prototype using 3D printing with PLA material and conducted grasping experiments. The control system includes a stepper motor (42HS4148), a microcontroller (STM32F103), and a driver (TB6600). We tested the end effector on various objects: five types of fine bobbins (end diameters 18–23 mm), two types of coarse bobbins (35 mm and 47 mm), and a tennis ball (66 mm diameter). Each test involved four phases: (1) approaching and lifting the object 5 cm, (2) accelerating horizontally at $10\,\text{mm/s}^2$ for 3 s, (3) moving at constant speed $30\,\text{mm/s}$ for 1 s, and (4) decelerating at $15\,\text{mm/s}^2$ for 2 s. Success is recorded if the object does not fall during a phase. Each object underwent 30 trials, and results are averaged. The end effector demonstrated high adaptability and stability, as shown in Table 3.
| Target Object | End Diameter (mm) | Mass (g) | Success Rate by Phase (%) | Average Success Rate (%) |
|---|---|---|---|---|
| Fine Bobbins | 18 | 110 | 90.00, 86.67, 86.67, 83.33 | 86.67 |
| 18 | 60 | 90.00, 86.67, 86.67, 86.67 | 87.50 | |
| 20 | 78 | 93.33, 93.33, 93.33, 90.00 | 92.50 | |
| 21 | 68 | 93.33, 90.00, 90.00, 86.67 | 90.00 | |
| 23 | 67 | 93.33, 90.00, 90.00, 90.00 | 90.83 | |
| Overall Average for Fine Bobbins | 87.33 | |||
| Coarse Bobbins | 35 | 23 | 93.33, 90.00, 90.00, 90.00 | 90.83 |
| 47 | 57 | 96.67, 96.67, 96.67, 93.33 | 95.83 | |
| Overall Average for Coarse Bobbins | 91.67 | |||
| Tennis Ball | 66 | 60 | 93.33, 90.00, 90.00, 90.00 | 90.00 |
The results indicate that the end effector achieves an average success rate of 87.33% for fine bobbins, 91.67% for coarse bobbins, and 90.00% for the tennis ball. Success rates generally increase with object diameter, due to better force distribution and geometric constraints. Most failures occurred in Phase 1, attributed to misalignment between the end effector and object centers; in industrial settings, positioning guides would mitigate this. The end effector also successfully performed enveloping grasps on a large yarn cylinder (diameter 110 mm, mass 750 g), confirming its versatility. The passive compliance from spring configuration allowed adaptive grasping without external sensing, highlighting the end effector’s robustness.
In terms of force capacity, the end effector’s fingertip contact force $F_3$ can reach up to 8 N theoretically. From the force balance equation for static equilibrium:
$$G = 3 F_1 \mu \cos \sigma$$
where $G$ is object weight, $F_1$ is fingertip force, $\mu = 0.6$ is friction coefficient, and $\sigma$ is the angle between force and horizontal (near zero). For $F_1 \approx 8\,\text{N}$, the end effector can grasp objects up to $G = 3 \times 8 \times 0.6 \times 1 = 14.4\,\text{N}$, or about 1.47 kg, far exceeding the tested bobbin weights (0.023–0.11 kg). Thus, the end effector provides ample gripping force for textile applications.
Our design and analysis underscore the importance of biomechanical inspiration in developing effective end effectors. The tendon-driven approach, combined with optimized spring stiffness, enables passive adaptability that simplifies control and enhances reliability. The stability model offers a quantitative framework for assessing multi-finger grasps in three dimensions, which can inform future end effector designs. Limitations include sensitivity to initial alignment and wear on tendon mechanisms over time; future work could incorporate sensors for active feedback or explore alternative materials for durability.
In conclusion, we have presented a comprehensive design and analysis of a bionic end effector finger for bobbin grasping. This end effector features an underactuated tendon-pulley system with passive spring compliance, allowing it to adaptively switch between fingertip and enveloping grasps. Through rigorous modeling of gripping forces, spring optimization, and stability analysis, we have tailored the end effector for multi-size bobbins in textile automation. Experimental validation with a 3D-printed prototype demonstrates high success rates and robustness, affirming the end effector’s practical utility. This work contributes to the advancement of robotic end effectors in industrial settings, emphasizing the value of bioinspired principles for adaptive and stable manipulation.