Design of a bidirectional compliant gripper for robotic object manipulation

Abstract

This study presents the design and development of a monolithic compliant gripper with bidirectional actuation, enabling both active grasping and full release within a single structural unit. To address these directional constraints commonly observed in conventional compliant mechanisms, the design incorporates a flexure-based cartwheel hinge that provides symmetric deformation and enhanced orientation adaptability. The gripper geometry is obtained through topology optimization to maximize material efficiency and mechanical advantage. Finite-element analysis and experimental validation confirm that the 3D-printed structure (30 g) achieves substantial bidirectional displacement under a 5 N input force. Performance testing demonstrates stable grasping, with successful manipulation of cubic objects at inclination angles up to 40° and pyramidal objects up to 50°. The integration of the cartwheel hinge enables more pronounced bidirectional motion and increased rotational capability, with a hinge thickness of 0.3 mm allowing rotation exceeding 90°, thereby improving grasping adaptability. Additionally, the gripper employs a modular structural design that allows rapid scaling from a two-finger pinch to a four-finger wrap. This system provides a lightweight and versatile solution for complex robotic manipulation tasks without the need for hardware redesign.

Introduction

A key challenge in gripper design is developing versatile mechanisms capable of handling a wide range of objects within a single operation. One significant difficulty lies in adapting to dynamic environments; for instance, the same object may need to be grasped from different orientations owing to placement constraints.^1,2 Soft mechanisms offer enhanced adaptability and minimize the risk of damaging delicate objects. However, excessively soft materials may reduce gripping force, limiting their effectiveness. ³ To address these limitations, compliant grippers have emerged as a promising solution, utilizing flexible structural elements to achieve controlled deformation. By integrating material and structural design, compliant grippers can conform to the shape of objects and apply appropriate force during handling. This adaptability not only reduces the risk of damage to fragile items but also improves safety in human-centered environments.^4,5 As a result, compliant grippers stand out as highly effective tools for complex manipulation tasks, including medical applications. ⁶ Depending on the intended application, compliant grippers can be designed in various forms, such as finger-like or claw-like structures, with further refinements made by adjusting the number of fingers.^7,8

Various methodologies have been explored to develop compliant grippers capable of multi-angle interaction. For instance, Hussain et al. ⁹ utilized screw theory for kinematic modeling and design, while Du et al. ¹⁰ employed topology optimization subject to stress constraints. As a core design methodology often supported by finite-element analysis (FEA), topology optimization begins with a defined 2D or 3D geometric model. ¹⁰ This process balances flexibility with structural strength by optimizing mass distribution to meet specific performance criteria. Furthermore, it enhances structural integrity and optimizes internal force pathways,^11–13 ensuring the gripper remains reliable under diverse loading conditions.^12,14 For large deformations, optimization challenges and limitations arise when linear material models are used instead of more suitable nonlinear materials. ¹⁵ In addition, additive manufacturing (3D printing) provides a feasible fabrication method; however, trade-offs exist between design and real-world performance due to material anisotropy and printing resolution. ¹⁶

The adaptability of compliant grippers is fundamentally rooted in their flexible structures and specialized materials, which allow them to conform seamlessly to diverse object geometries. ¹⁷ By utilizing deformable materials such as silicone, soft grippers provide the exceptional compliance necessary for damage-free handling of fragile or irregularly shaped items.^5,18 Beyond material properties, functional versatility can also be enhanced through integrated technologies, including electroadhesion, which enable secure manipulation with minimal applied force. ¹⁹ Similarly, innovations in compressible mechanisms allow for advanced shape adaptation, ensuring the gentle handling required in precision-driven fields such as manufacturing and biomedical robotics. ²⁰

By enabling independent finger motion, multi-degree-of-freedom (DOF) grippers such as DexGrip, which combine a suction palm with rotational grasping surfaces, provide the dexterity required for complex in-hand manipulation.^5,21,22 The design of these systems can be further enhanced by topology optimization, which balances weight reduction with structural strength to facilitate versatile grasping in dynamic environments.^11,18,23 However, this increased mechanical complexity is not without cost. The added dexterity introduces control challenges and higher computational overhead. Consequently, there is a persistent trade-off between high-level functionality and the economic or technical feasibility of the system.^23,24

To improve the flexibility and accuracy of compliant systems, flexure hinges such as circular flexure hinges have been applied.^25,26,27 Liu et al. ²⁸ demonstrated the use of a flexure hinge in a compliant prosthetic finger. The ability of such mechanisms to conform to diverse object geometries improves grip stability without the need for precise motor control and feedback systems.^29,23 In addition, the monolithic construction of compliant grippers reduces energy loss at joints and simplifies the manufacturing process. These attributes make them particularly well-suited for applications in industrial automation, prosthetics, and robotic manipulation, where efficiency, adaptability, and ease of production are essential. Despite these advances, many existing compliant gripper designs still face limitations in achieving versatile actuation and modular adaptability within a single integrated structure. To address this challenge, a bidirectional modular compliant gripper is developed using topology optimization and flexure hinge design to enhance grasping adaptability. Compared with existing compliant grippers and topology-optimized grasping mechanisms, the proposed design introduces several key advancements. Conventional compliant grippers often exhibit limited release efficiency due to unidirectional actuation, resulting in incomplete structural recovery. In contrast, the proposed gripper integrates a cartwheel flexure hinge that enables effective bidirectional actuation, significantly improving release performance and allowing the fingers to return closer to their original configuration.^30,31 In addition, the modular configuration enables rapid adaptation to different grasping tasks, ensuring versatility across a wide range of object geometries while maintaining monolithic structural integrity.

This work is divided into two primary sections. The first section presents the design specifications, topology optimization process, and integration of flexure hinges for performance enhancement. The second section focuses on the development and validation of the compliant gripper, including grasping evaluations and the implementation and modular configuration.

Materials and methods

The design of the compliant gripper was driven by the requirement for structural adaptability to specific object geometries and force profiles. A target object envelope of 50 × 40 mm was established, utilizing a fundamental two-finger configuration as the design baseline.

Material

To support large-deflection motion, thermoplastic polyurethane (TPU) was selected as the substrate material. TPU provides a balance of flexural elasticity and compatibility with 3D printing, enabling the fabrication of complex, monolithic compliant structures. TPU was characterized by an elastic modulus of 2.7 × 10⁶ Pa and a Poisson's ratio of 0.45. To ensure the reliability of the numerical simulations, mechanical characterization of the 3D-printed TPU specimens was performed under the same printing conditions as those used for the gripper fabrication. All tensile specimens were printed with 100% infill to eliminate internal porosity effects and to obtain intrinsic material properties. To evaluate the anisotropic mechanical behavior induced by the fused filament fabrication process, three printing orientations were investigated. The XY orientation corresponds to the conventional raster pattern printed in alternating transverse layers on the build plate. The YZ orientation represents vertically stacked layers along the build direction, while the Z orientation corresponds to specimens printed upright along the vertical axis. Uniaxial tensile testing was conducted in accordance with ISO 37 for thermoplastic elastomers. In Table 1, the measured tensile properties are summarized. Among the tested orientations, the XY specimens exhibited the highest elastic modulus and tensile strength. Since the gripper was fabricated in the XY printing orientation, the corresponding material properties were adopted in the finite-element simulations. The higher stiffness and tensile strength of the XY orientation provide improved load transfer capability from the input mechanism to the compliant structure and enhance resistance to tearing under large deformation.

Table 1.

Tensile properties of TPU material.

	Typical value
	XY	YZ	Z
Modulus (MPa)	2.76 ± 0.2	2.20 ± 0.2	0.81 ± 0.2
Tensile strength (MPa)	34.44	28.18	23.93
% Elongation	>400%	>690%	>600%

As shown in Figure 1, stress–strain curves of the TPU specimens printed in the XY orientation were obtained from three samples. Although the material exhibits hyperelastic characteristics at large strains, the present FE simulations were conducted using a linear elastic material model. Geometric nonlinearity was included in the analysis to account for large deformation effects. Within the deformation range observed in the compliant joint during gripper operation, the strain level remained predominantly within the initial quasi-linear region of the stress–strain curve. Therefore, the linear approximation was considered sufficient to capture the global displacement behavior and load transfer mechanism of the structure. The use of experimentally measured material properties under identical printing conditions ensures that the numerical model reflects the actual mechanical response of the fabricated gripper, thereby improving simulation reliability.

Figure 1.

Stress–strain behavior of TPU material.

Design domain

To design the compliant gripper, the initial configuration is defined as a monolithic structure capable of securely manipulating two target geometries: a 40 mm cube and a 40 mm diameter sphere. The design domain is specified as a 160 × 140 mm rectangular plane, with a 50 × 40 mm central void reserved to provide functional clearance and represent the primary grasping envelope. The design process begins with this 2D domain, where boundary conditions, including fixed regions and input ports, are defined according to the desired actuation behavior (Figure 2(a) and (b)).

Figure 2.

(a) Initial dimensions for topology optimization and (b) boundary conditions.

This boundary condition ensures that the generated material distribution remains outside the immediate contact zone until the intended actuation occurs. In Figure 2, the specific layout, including fixed constraints and load application points used to drive the optimization, is illustrated. In Figure 2(b), the input and output ports within the initial design domain are defined to establish the boundary conditions for the topology optimization process. Roller constraints are applied at specific locations to control the deformation of the structure while allowing motion along designated directions. These constraints are oriented in both the same and opposing directions relative to the applied forces, enabling the optimized structure to deform along both the vertical and horizontal axes. The applied loads are modeled as concentrated forces acting on selected surface nodes. Each input port is subjected to a 5 N force representing the actuation input in the topology optimization framework. A dummy force of 2.5 N is applied at each output port to guide the deformation behavior during the optimization process. This dummy force defines the intended motion direction and ensures consistent force redistribution within the compliant structure. When a 5 N force is applied simultaneously to both input arms, the output ports move toward each other, enabling the gripper to securely grasp an object. This behavior is referred to as a dummy force. ¹⁷

Topology optimization

In the past, the design of compliant grippers has involved a significant trade-off between structural flexibility and gripping force. To address this issue, the present study employs a topology optimization framework aimed at minimizing global compliance (maximizing stiffness) while satisfying strict volume and stress constraints. The objective is to optimize the internal force transmission of the mechanism. By balancing the stiffness of the compliant members, the design ensures that the input force is efficiently transmitted to the gripping interface, providing sufficient pressure to securely manipulate objects without structural failure.

Bendsøe and Sigmund laid the groundwork for modern topology optimization methods, particularly the solid isotropic material with penalization (SIMP) technique.^32,33

Figure 3 present the mesh convergence study used to evaluate the effect of element discretization on the numerical results. The model was analyzed with progressively refined meshes, showing no significant differences in deformation patterns or output displacement. Therefore, a uniform grid size of $2\mathrm{mm}\times 2\mathrm{mm}$ was selected to balance computational efficiency and solution accuracy in the 2D simulations. The objective function for the optimization problem was then defined.

Figure 3.

Mesh convergence study based on boundary conditions.

Figure 4 illustrates the overall topology optimization process. The optimization problem is formulated as a compliance minimization problem.

Figure 4.

Topology optimization process.

The optimization framework employed in Abaqus minimizes structural compliance while satisfying constraints on both volume and stress through the SIMP method. The optimization problem is formulated by defining an objective function that minimizes structural compliance under the specified boundary conditions:

$$\underset{\rho }{min}C(\rho )=f^{T}u$$

(1)

where ρ determines the optimal distribution of material density such that the resulting structure achieves the optimization objective. The term C(ρ) refers to the structural compliance, which is computed as the dot product of the externally applied force vector f and the resulting displacement vector u. The applied forces at the designated input and output ports define the intended actuation mode of the mechanism. By minimizing C(ρ) under these prescribed loading conditions, the optimization process determines a material distribution that efficiently transfers forces along the specified transmission path. Consequently, the obtained topology deforms according to the intended motion pattern and serves as the structural basis for the compliant gripper. The resulting displacement u is obtained by solving the equilibrium equation:

$$K(\rho )u=F$$

(2)

where K(ρ) is the stiffness matrix of the structure, which varies according to the elementwise material density ρ. Equation (2) describes the balance between internal stiffness and external loading in FEA. This equilibrium equation is solved iteratively at each step of the optimization process to account for changes in the material layout. The SIMP interpolation model governs the relationship between material density and stiffness:

$$E(\rho )={\rho }^p{E}_0$$

(3)

Where the Young's modulus E(ρ) of each element is expressed as a function of the material density, which is raised to a penalization power p. In this study, the reference modulus of the TPU filament is defined as $E_0$ = 2.7 MPa. The penalization factor p, set to 3, is used to promote binary (0 or 1) material distributions by discouraging intermediate densities. To ensure material efficiency, a volume constraint is introduced:

$$\underset{\mathrm{\Omega }}{\int }\rho (x)d\mathrm{\Omega }\le \alpha V_0$$

(4)

$$\sigma (x)\le {\sigma }_{max}$$

(5)

where $V_0$ is the total volume of the design domain, and α ∈ (0,1] is the allowable volume fraction. The volume fraction α is set to 0.2, meaning that the optimized structure is restricted to using no more than 20% of the total design space. To prevent mechanical failure under loading, a stress constraint is applied. This constraint ensures that the von Mises stress σ(x) does not exceed the maximum allowable stress ${\sigma }_{max}$ at any point.

The density vector and stiffness matrix are defined as functions of the material density distribution, where each element is associated with a density variable ${\rho }_e$ . The sensitivity of compliance with respect to the element density can be expressed as:

$${\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }\rho }=-p{\rho }_e^{p-1}(f^{T}u)}$$

(6)

To improve the optimization process, the density variable is transformed into an unconstrained space while preserving monotonicity. In this study, the following transformation is adopted:

$${\rho }_e={\displaystyle \frac{1}{1+e^{-x_e}}}$$

(7)

The design variable $x_e$ is defined in an unbounded domain, thereby removing the explicit bound constraints during optimization.

In addition to evaluating the objective and constraint functions, a gradient filtering technique is applied to the sensitivity field. In this approach, the sensitivity of each element is replaced by a weighted average of the sensitivities of its neighboring elements within a specified filter radius, resulting in a smoother and physically meaningful material distribution. The sensitivity with respect to the transformed design variable is expressed as:

$${\displaystyle \frac{\mathrm{\partial }C}{\mathrm{\partial }\rho }{\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }{x}_e}=-p(f^{T}u)\left({\displaystyle \frac{1}{{(1+e^{-x_e})}^{p+1}{e}^{x_e}}}\right)}}$$

(8)

For nonlinear constraints, optimization is performed using the direct gradient projection method. The gradients of both the objective and constraint functions are used to update the design variable, ensuring stable convergence.

Optimization proceeds iteratively until the constraints are satisfied, converging after 32 iterations. The convergence history of the objective function and volume fraction is shown in Figure 5.

Figure 5.

Objective function and volume fraction during compliance minimization of the compliant gripper.

After performing sensitivity analysis, the design variables were updated accordingly. The process continued until the objective function converged. The optimized structure is shown in Figure 6. The model was designed such that when a force is applied at the input port, motion occurs within the structure. The displacement from the original position at the input port is defined as $\delta d_{in}$ , whereas the displacement at the fingertip region is defined as $\delta d_{out}$ . The resulting displacement at the output port is sufficient to move the structure into a grasping position. However, this initial design exhibited limitations in gripping performance. Specifically, when attempting to achieve maximum gripping force, excessive deformation was observed in Figure 6. As a result, the structural integrity of the optimized model was compromised. This issue necessitated further design modifications to ensure that the gripper could achieve maximum output displacement without compromising structural performance.

Figure 6.

Optimized structure and deformation visualization.

Enhancing the optimized model

To investigate the cause of structural failure, the model's stress and displacement distributions were analyzed using FEA. The results indicated that the highest stress concentrations occurred at the joint between the center section and the finger, as well as at the upper structure connecting the finger to the gripper body. The selected approach involved integrating a flexure hinge to increase flexibility at designated pivot points. The hinge acts as a stress-relief region when a concentrated force is applied at the input port. Elliptical and rectangular flexure hinges were chosen, each serving a different function. The elliptical hinge was designed to reduce stress concentration at the hinge location while enhancing flexibility, allowing smoother motion. This hinge type was strategically placed at the upper section of the optimized model.

A common limitation of compliant grippers is their low release efficiency. To address this issue, flexure hinges are introduced to enhance the gripper's ability to recover its original shape. In the optimized model, the movable joint beneath the fingers, which connects them to the gripper body, exhibited limited movement due to excessive deformation. Modifications were made to ensure that the joint effectively transfers force from the input port to the fingertips while also maximizing its ability to function as a rotational axis. The greater the rotational capability at this location, the higher the output displacement, thereby improving the overall gripping performance. To achieve this, a rectangular flexure hinge was selected because it allows greater angular rotation. As shown in Figure 7, this hinge is strategically placed at the lower joint to enhance rotational movement. Finally, ribs were added to the central section to prevent excessive bending.

Figure 7.

Hinge solutions: (a) elliptical and (b) rectangular.

After incorporating the hinges, the maximum output displacement of the gripper no longer exhibited excessive deformation. This outcome demonstrates that integrating flexure hinges into the optimized model effectively redistributes stress concentration toward the hinge locations, thus preventing deformation in other areas. Additionally, the use of flexure hinges enhances the overall flexibility of the gripper's movement, allowing smoother structural motion. Enhancing the gripper's ability to release its fingers is essential for improving the grasping mechanism. Previously, the gripper was operated by applying a pushing force at the input port to initiate the gripping motion. However, for the release action, a pulling force in the opposite direction was introduced to retract the fingers and open the gripper. This concept is illustrated in the release mechanism integrated with the hinge model, where the gripper successfully releases its model with cartwheel hinges during pus fingers. While the current modification enables finger release, it is still insufficient to return the fingers to a fully flat position. The next step aims to refine this mechanism so that the gripper's fingers can fully extend and achieve full contact with the grasped surface.

During the release action, challenges in achieving full release were identified, particularly due to the added hinge mechanism along the central axis. Moreover, the elliptical and rectangular hinges were unable to facilitate bidirectional motion. As such, the flexure hinges in the original model were unable to accommodate both pull and push actions, limiting the overall effectiveness of the release mechanism. To resolve this limitation, a new flexure hinge design was required that could withstand both pulling and pushing forces without compromising the gripper's performance. Therefore, a cartwheel flexure hinge was chosen because it could support bidirectional motion while preventing excessive deformation.

In Figure 8(a), the hinge replacement was guided by the positions of the original joints. To facilitate vertical motion under both pulling and pushing forces, a hinge was integrated at the lower joint between the finger and the base. In addition, the rotational joint near the fingertip was modified to enable sufficient angular displacement, allowing the fingers to fully open in the release configuration. The required angular range was calculated through geometric analysis, with the relevant dimensions presented in Figure 8(b), and the associated equation is given as follows ²⁵ :

$${\theta }_{Zmax}={\displaystyle \frac{R}{Et\cdot SF}{\sigma }_y}$$

(9)

Figure 8.

(a) Cartwheel hinge design, (b) optimized model with cartwheel hinges during pushing, and (c) optimized model with cartwheel hinges during pulling.

where R represents the distance from the center point to the base of the hinge, t is the hinge thickness, b is the base length, and ${\theta }_{zmax}$ is the maximum rotational angle of the hinge. The equation calculates the maximum possible rotation angle, incorporating a safety factor (SF), which is set to 20 in this study. The gripper was fabricated using 3D printing with a TPU filament, characterized by a Young's modulus of 2.7 MPa and a yield strength of 9 MPa. As shown in Figure 8(c), to achieve full finger release in a flat position, the cartwheel hinge must rotate at least 30°.

In Figure 8(b), a cartwheel hinge with dimensions R = 5 mm and t = 1 mm is implemented. The addition of the cartwheel hinge greatly improves the grasping and releasing capabilities of the gripper. FEA results indicate that excessive deformation mainly occurs at the upper connection and the central section of the gripper. To reduce deformation, the upper section was modified using an elliptical flexure hinge to improve stress distribution. The revised design also incorporates an additional cartwheel hinge (Figure 8(b) and (c)). The effectiveness of this modification was evaluated through both FEA and experimental testing using a 3D-printed prototype. The results show that the modified design successfully eliminates excessive deformation, allowing the gripper to release its fingers into a flat position. This modification resolves the issue. Further changes to the central section were not required.

The hinge behavior is first estimated using an analytical model. As shown in Figure 9, a hinge thickness of 0.3 mm yields a rotation angle of approximately 100°, while experimental measurements indicate 90–95°. As the hinge thickness decreases, the rotation angle increases, following a consistent trend in both simulation and experiment. Although minor deviations occur at smaller thickness values, the overall behavior remains consistent. This confirms that hinge performance can be effectively controlled by thickness and that the proposed model provides a reliable design reference.

Figure 9.

Relationship between hinge thickness and maximum rotation angle.

Results confirm that the cartwheel hinge was the optimal choice, as it accommodated both push and pull actions without causing excessive deformation. Subsequently, the design was rescaled to incorporate a four-finger configuration, enabling multi-DOF grasping. One scenario involved a four-finger gripper where each finger could be activated based on the object's orientation. For instance, when picking up an object next to a wall or in a confined space, the gripper could use only two fingers to grasp the object, while the remaining two fingers remained disengaged.

Prototype development

Figure 10 shows the compliant gripper with a symmetric two-finger configuration and integrated flexure hinges, enabling deformation-based grasping and releasing. The overall frame measures 140 mm in both width and height, with a finger spacing of 40 mm to accommodate objects of different sizes. The gripper was fabricated using a FlashForge Guider 3 Ultra 3D printer with 100% infill and a thickness of 5 mm.

Figure 10.

Compliant gripper model and dimensions.

As illustrated in Figure 10, two types of cartwheel-style flexure hinges were integrated into the gripper design (Hinge A and Hinge B). Both types have a radius of 5 mm and a span length of 10 mm. They serve the same purpose, but their ligament thicknesses differ and are defined as $t_1$ and $t_2$ . The first hinge type (Hinge A), located at the top of the gripper, has dimensions $R_1=5$ mm, $t_1=1$ mm, and is located at four positions along the upper frame. Based on equation (6), this hinge configuration yields a maximum out-of-plane angular deflection ${\theta }_{1\max }$ of 30°. The second hinge type (Hinge B) is applied at the lower joints of the fingers and features a significantly thinner section, defined by $R_2=5$ mm and thickness $t_2=0.3$ mm. This design increases flexibility, allowing the fingers to bend further inward. The calculated ${\theta }_{2\max }$ for this hinge is 100°. The geometric parameters and corresponding estimated maximum rotation angles of both hinge types are summarized in Table 2. The difference in thickness between the two hinge types reflects their distinct roles in motion. Thinner hinges facilitate greater compliant motion while thicker hinges enhance structural resilience. This configuration, which includes hinge geometries and overall dimensions, serves as the reference model for subsequent simulation and experimental analysis.

Table 2.

Geometric parameters of the cartwheel hinges.

Parameter	Hinge A	Hinge B
Radius (mm)	5	5
Thickness (mm)	1	0.3
Span (mm)	10	10
Maximum angle (degree)	$30\circ \le {\theta }_1\le 150\circ$	$-24\circ \le {\theta }_2\le 100\circ$

The range of motion (ROM) is a crucial factor in determining a gripper's adaptability to objects of different sizes and shapes. Compared with rigid grippers constrained by fixed joint angles, compliant grippers achieve a wider ROM through material deformation. The performance of the compliant gripper is assessed through motion path analysis. This method enables the analysis of gripper deformation during actuation using an Intel RealSense D435i depth camera. The acquired images are processed using the open-source software ImageJ to extract displacement data and construct motion path graphs.

As shown in Figure 11, the compliant gripper can perform both pushing and pulling actions. These motions demonstrate the adaptability of the gripper design. Figure 12 shows the motion paths of the pushing and pulling phases, highlighting the different deformation behaviors in each direction. This result confirms that the gripper can change shape and respond effectively to both types of actuation.

Figure 11.

Motion paths of the gripper during actuation: (a) pushing action and (b) pulling action.

Figure 12.

Experimental capture of the gripper's compliant motion.

The functionality of the compliant gripper is primarily governed by the application of input displacement or force at designated input ports. This input induces a controlled deformation in the structure, resulting in the desired output motion at the gripper tips. When a vertical displacement or compressive force is applied at the input ports, the internal compliant segments, mainly the flexure hinges, bend. This deformation propagates through the structure, producing outward or inward motion at the output ports depending on the direction of the applied input. Consequently, the gripper can perform grasping or releasing actions. When the input port is displaced downward (pushing), the fingers move inward to grasp the object. Conversely, when the input is pulled upward (pulling), the gripper returns to its original configuration, allowing the object to be released. The deformation behavior of the structure is designed to ensure repeatability and sufficient output displacement for stable contact during grasping.

Experiments

The compliant gripper operates in two primary modes: grasping and releasing. Grasping is achieved by applying force or displacement at the input ports, causing the compliant mechanism to deform and drive the fingers to close around an object. The release action occurs when the arms of the gripper are pulled toward each other, passively reconfiguring the structure and allowing the fingers to open. To experimentally verify the gripper's performance, a motion capture setup was constructed. The setup consisted of a linear actuator used to generate input displacement for both pushing and pulling actions. The motion of the gripper was recorded using an Intel RealSense D435i depth camera. The recorded images were analyzed using the open-source software ImageJ to track key marker positions and convert them into motion path data.

As shown in Figure 12, the experiment was conducted using a four-finger gripper. To introduce additional DOFs to the system, the number of fingers was increased. This enhancement enabled the gripper to handle a wider range of objects. The entire system was actuated by a runner connected to a threaded rod driven by a stepper motor. To execute the pushing and pulling actions, the motor moves the runner upward or downward, and the analysis is based on the input displacement $\delta d_{in}$ . During the pushing action, displacement is measured at the input ports. Likewise, during the pulling action, displacement is measured at the output ports. The analysis is divided into two parts. The first part examines the displacement relationship by observing the gripper's response to the applied input. The second part evaluates the gripper's ability to grasp a range of objects using the four-finger configuration, including objects not considered during the pre-optimization phase. This assessment demonstrates how the new hinge design and the increased number of fingers contribute to the overall improvement in performance.

To evaluate the gripper's adaptability under different grasping conditions, several test objects with different geometries were selected. Gripper orientation testing was performed using three object geometries: spheres, cubes, and pyramids. All objects had a characteristic dimension of 40 mm. Specifically, the sphere had a diameter of 40 mm. The cube measured 40 × 40 × 40 mm, and the pyramid had a square base of 40 × 40 mm with a height of 40 mm. To ensure consistency across trials, the position and orientation of each object remained constant throughout the experiments.

Results and discussion

Two-finger test

During gripper operation, the motion path provides insight into both the input and output displacements. During the release phase, the motion involves displacement in both the X and Y directions, forming a curved trajectory. As a result, plotting a conventional displacement relationship becomes less intuitive because of the multidirectional nature of movement. In contrast, during the pushing phase, when compressive displacement is applied at the input ports, the direction of motion can be clearly defined. In this study, input displacement is measured along the vertical axis, corresponding to downward actuation. Output displacement is observed along the horizontal axis as the gripper arms move inward to grasp the object. Figure 13 shows the displacement relationship of the pushing action. The experimental results are compared with FEA results for both pushing and pulling actions.

Figure 13.

Input and output displacement relationship during (a) pushing action and (b) pulling action.

The simulation results closely match the experimental data. Minor deviations at low and high displacement levels may result from material compliance, mechanical backlash, or unmodeled nonlinearities in the physical setup. Overall, the comparison validates the simulation model and confirms that the designed compliant gripper performs effectively under the targeted actuation conditions. To quantitatively evaluate the agreement between simulation and experiment, error metrics were calculated for both pushing and pulling configurations. For the pushing action, the root-mean-square error (RMSE) between the simulated and experimental displacements was 0.07 mm ± 0.14 mm. The slightly higher deviation observed at the initial stage of pushing is attributed to the onset of motion in the cartwheel hinge mechanism. At small input displacements, minor mechanical compliance and initial clearance effects can introduce small offsets before the hinge fully engages in stable deformation. For the pulling action, the RMSE was approximately 0.09 mm ± 0.25 mm. The close agreement between simulated and experimental results is attributed to the integration of flexure hinges, which localize deformation and improve structural predictability. This refinement reduces the gap between the numerical model and the physical prototype. At larger displacements, discrepancies increase due to geometric nonlinearity in the TPU structure, amplified deformation in thinner hinge sections, and the growing influence of fabrication tolerances and mechanical gaps.

Despite these deviations, the overall displacement trends between simulation and experiment remain consistent across the entire motion range. The gripper maintains stable bidirectional functionality, successfully performing both pushing and pulling motions in accordance with the proposed design concept. Furthermore, the geometric advantage (GA), defined as the ratio of output displacement to input displacement, was determined for both the simulation and experimental cases. The simulated GA remained constant at 1.02, indicating consistent motion transmission. The experimental GA showed higher values at the beginning due to the low resistance in the hinges at the start of deformation. As displacement increased, the experimental GA gradually converged toward the simulation values. Throughout the motion range, the GA remained greater than 1, confirming that the mechanism produced a larger output displacement than the input. This behavior indicates effective displacement amplification arising from the optimized compliant geometry. By promoting efficient force transmission between the input and output under the prescribed loading configuration, the topology optimization process forms compliant regions that function in a lever-like manner. Consequently, a GA greater than unity arises primarily from geometric transformation of motion rather than from material softening.

Four-finger test

To increase the contact area during manipulation, the original two-finger design was expanded into a four-finger compliant gripper module. This configuration significantly increases both contact stability and task versatility. The performance of the four-finger system was validated through an orientation-based grasping test using a top-down approach. During these trials, the gripper initially approached the target vertically; while the object remained in a fixed position, the gripper's orientation was incrementally rotated through various angles to test the limits of its workspace. Figure 14(a) presents a heatmap showing the success rate for the sphere when using the four-finger compliant gripper. Regardless of the orientation angle in the top-down configuration, the success rate was highest when the gripper was positioned near the center of the sphere. The success rate gradually decreased as the contact region moved away from the center. This finding indicates that, for spherical objects, minimal orientation adjustment is required if contact occurs within an appropriate central region.

Figure 14.

Operational angle range of the gripper: (a) grasp on the sphere, (b) failed grasp on the cube, (c) angular operation range during cube grasping, and (d) angular operation range for pyramid grasping.

In Figure 14(b), the gripper fails to grasp the cube due to the loss of contact symmetry. When the gripper's fingers move away from the cube's center, the grasp becomes unstable and ineffective. This behavior highlights the importance of balanced contact forces in maintaining grasp stability. Therefore, when the gripper's orientation shifts into this configuration, the grasping success rate effectively decreases to zero, indicating a clear operational boundary. Figure 14(c) and (d) illustrates the operational angle ranges of the gripper when grasping the cube and pyramid, respectively. For the cube, the gripper successfully grasps the object at angles up to 40° from the initial top-down direction. Beyond this point, the gripper enters the unstable configuration shown in Figure 14(b), resulting in grasp failure. In the case of the pyramid, the gripper could not successfully grasp the object when operating in a pure top-down orientation. The gripper's fingers do not move strictly inward; instead, their motion path includes a slight upward component. Consequently, stable grasping was only achieved between 30° and 80°, resulting in an effective grasping range of 50°.

Under a top-down perspective, the fingers may contact the ground before completing a grasp. This limitation arises from the approach angle associated with the top-down configuration. To explore the gripper's ability to function under a broader range of orientations, additional tests were conducted using a side-approach strategy (as shown in Figure 15(a)). Figure 15(b) shows the range of approach angles where the gripper successfully grasps the object. Compared to the top-down approach, the side approach demonstrated greater effectiveness, especially when handling objects such as cubes. In contrast, Figure 15(c) shows a decrease in grasping performance when the gripper applies the same side approach to a pyramid-shaped object. This observation suggests that while the side approach increases the usable orientation range, its success still depends on the shape of the object. The present study evaluates objects with a characteristic dimension of 40 mm under controlled positional and orientation conditions. Therefore, the results demonstrate bidirectional actuation feasibility rather than universal manipulation capability.

Figure 15.

Operational angle range of the gripper: (a) gripper orientation during side-approach, (b) angular range of operation for cube grasping, and (c) angular range of operation for pyramid grasping.

Object grasping evaluation

To evaluate the performance of the developed gripper, grasping tests were conducted using real objects under varying conditions. Ten objects with diverse physical characteristics were selected for testing: a stapler, a screw, a pen, a screwdriver, a keychain, a USB drive, a coin, a glue stick, a cube, and a sphere (Figure 16). These objects varied in size, shape, texture, and stability to represent real-world variability. Among them, four representative objects (a cube, a sphere, a glue stick, and a keychain) were selected for detailed analysis, as they represent different geometric conditions for evaluating the adaptability of the gripper.

Figure 16.

Test objects used in the experiment: screwdriver, pen, sphere, screw, cube, keychain, USB drive, coin, glue stick, and stapler.

The tests were divided into three grasping orientations: vertical, horizontal, and tilted grasping, as shown in Figure 17. In each orientation, the gripper attempted to hold each object ten times. The number of successful and unsuccessful attempts was recorded to evaluate grasping reliability under each directional condition.

Figure 17.

Grasping of objects under different orientations: (a) vertical, (b) tilted, and (c) horizontal.

In Table 3, the experimental results are summarized. The ten objects were selected to evaluate whether the gripper could successfully grasp items under the constraints defined during the optimization. The objects were limited to cross-sectional dimensions not exceeding 40 × 50 mm and represented a variety of shapes and geometries. For instance, a cube and a sphere were chosen to represent basic shapes. A USB drive was used as a rectangular prism due to its appropriate size. For cylindrical shapes, a glue stick and a pen were selected, differing in size and length. Additionally, irregularly shaped objects such as a keychain and a screwdriver were included to test the gripper's adaptability to non-uniform geometries. The results under different orientations reveal that the gripper can adapt to multiple grasping directions depending on object geometry. One limitation in the current prototype is the 5 mm finger thickness, which restricts its ability to lift heavier and bulkier objects. However, this relatively thin structure enhances accessibility in tighter spaces and improves grasping in alternative orientations. As shown in Figure 17(c), the gripper's fingers exhibited greater deflection when grasping the cube, allowing successful vertical and horizontal grasping. Nevertheless, performance in the tilted orientation was less effective for the cube due to the symmetric finger configuration, which prevented proper engagement with the object's corners. In contrast, objects smaller than the cube, such as the USB drive, could be grasped in the tilted orientation, as the fingers were able to hook or clamp onto the object and lift it. The coin, however, could not be lifted at all. As shown in the motion path, the gripper deflected upward along the Y-axis during actuation, leaving no effective contact with flat, thin objects. This behavior also applied to the screw, which could not be grasped in either tilted or horizontal orientations due to its limited height. Nonetheless, it was successfully lifted in the vertical orientation, where the fingers could directly engage with the screw's head. The pen also demonstrated reduced success in tilted and horizontal orientations; although it is long, its low height when lying flat limited its grasping performance. On the other hand, objects such as the sphere, stapler, glue stick, and screwdriver fit well within the gripper's grasping range and showed high success across all orientations.

Table 3.

Grasping the success rates of ten objects under vertical, tilted, and horizontal orientations.

Object	Grasping orientation
	Vertical	Tilted	Horizontal
Sphere	100%	100%	80%
Cube	100%	40%	90%
USB drive	100%	70%	100%
Keychain	20%	30%	70%
Coin	10%	-	-
Glue stick	100%	100%	100%
Screwdriver	100%	100%	100%
Screw	80%	-	-
Pen	90%	50%	20%
Stapler	100%	80%	70%

It is acknowledged that the gripper does not perform optimally under all situations. However, its ability to operate effectively across multiple orientations, enabled by topology optimization and hinge redesign, reveals its strong adaptability. For example, in scenarios where a cube is placed next to a wall and access from above is restricted, the gripper can still grasp the object successfully using a side approach. The gripper designed using topology optimization and improved with a cartwheel hinge was able to lift objects under predefined initial conditions. Consequently, it can accommodate a wider range of orientations. Nonetheless, limitations remain when dealing with flat or very small objects. Future design improvements may further enhance the gripper's adaptability and object-handling capabilities.

Conclusion

This paper presents a compliant gripper designed using topology optimization and flexure hinge integration to improve motion adaptability. Material properties of the 3D-printed structure were incorporated during optimization to reduce errors associated with printing resolution and material anisotropy. A prototype was fabricated and experimentally validated through motion tracking and force measurements, demonstrating both grasping (pushing) and releasing (pulling) actions. The RMSE between the simulated and experimental displacements was 0.07 mm ± 0.14 mm for pushing and approximately 0.09 mm ± 0.25 mm for pulling. These discrepancies are primarily attributed to hyperelastic material properties, which were simplified as linear elastic behavior in the numerical model to maintain computational efficiency. The design was further extended from a two-finger to a four-finger configuration, enabling the manipulation of objects with various shapes. Although limitations remain in fingertip force and release recovery for small flat objects, the gripper demonstrates strong potential for future compliant robotic manipulation systems.