Local Displacement Convex–Concave Gripper

• Local displacement convex–concave structure is defined by a sensorized pin-array that adapts to both convex and concave surfaces using local curvature measurements.
• The design leverages calibrated elastic responses and force-sensitive resistors to translate mechanical displacement into accurate 3D mapping and shape recognition.
• Empirical results show robust grasping and reliable terrain measurement on irregular surfaces by integrating frictional mechanics with local tactile feedback.

A local displacement convex–concave structure describes the geometric and mechanical relationship by which an array of discretized vertically compliant pins in a gripper prototype conforms to, measures, and grasps both convex and concave terrain patches. This structure is characterized by the local normal displacements imposed by the terrain’s curvature at each pin, the corresponding elastic force response, and the integrated tactile sensing required for shape recognition and 3D mapping. Developed for robust grasping in extreme and uncertain natural environments, such as cliffs and cave walls, the system achieves simultaneous adaptive grasping and terrain surface measurement, as demonstrated in the pin-array gripper described by Kato et al. (Kato et al., 13 Jan 2026).

1. Geometric Configuration of the Pin-Array System

The gripper is constructed with a pin-array arranged in a quasi-rectangular grid comprising 21 pins (3 blocks × 7 pins per block). The physical parameters are as follows: pin spacing along the X-axis (columns) is $X_{\text{pitch}}=14\,\mathrm{mm}$; along the Y-axis (rows), $Y_{\text{pitch}}=17.4\,\mathrm{mm}$. The gripper’s overall envelope is $150\,\mathrm{mm}$ (width) $\times$ $165\,\mathrm{mm}$ (height, fully extended) $\times$ $70\,\mathrm{mm}$ (depth). Pins consist of a polycarbonate shank, with the tip utilizing a standard brass nail ($d_{\text{pin}}\approx2\,\mathrm{mm}$), guided vertically via a dual-holder mechanism (fixed rear, slidable front) enabling controlled engagement with terrain surfaces.

This configuration supports spatially dense, point-wise measurement and interaction with both convex and concave geometries, independent of the macroscopic hand structure typical in conventional grippers. The local nature of displacement means each pin responds independently to substrate topography, an essential requirement for operation on rough, unknown terrain.

2. Mechanical Model: Displacement and Force Relations

For a local segment of smooth terrain parameterized as $z=z(x)$, the vertical displacement experienced by a pin at offset $x_i$ relative to the central axis is governed, to leading order, by the local curvature $Y_{\text{pitch}}=17.4\,\mathrm{mm}$0:

$Y_{\text{pitch}}=17.4\,\mathrm{mm}$1

Thus, for pin $Y_{\text{pitch}}=17.4\,\mathrm{mm}$2:

$Y_{\text{pitch}}=17.4\,\mathrm{mm}$3

where $Y_{\text{pitch}}=17.4\,\mathrm{mm}$4 is the nominal zero-contact level.

The elastic response of each pin is that of a cantilevered beam:

$Y_{\text{pitch}}=17.4\,\mathrm{mm}$5

with $Y_{\text{pitch}}=17.4\,\mathrm{mm}$6 denoting the Young’s modulus, $Y_{\text{pitch}}=17.4\,\mathrm{mm}$7 the second moment of area, and $Y_{\text{pitch}}=17.4\,\mathrm{mm}$8 the beam length. For the prototype regime, the linear regime dominates (experimentally, $Y_{\text{pitch}}=17.4\,\mathrm{mm}$9). For completeness, higher-order stiffness (e.g., $150\,\mathrm{mm}$0) is theoretically attainable but not observed in the primary operational regime.

Convexity ($150\,\mathrm{mm}$1) and concavity ($150\,\mathrm{mm}$2) enter only via the sign of $150\,\mathrm{mm}$3 and the number of pins engaged, with tighter curvature leading to larger displacements and forces.

3. Sensing Integration, Calibration, and Surface Reconstruction

Each pin is equipped with a force-sensitive resistor (FSR) measuring its individual compression. The calibration process for the displacement readings uses a two-point linear mapping (Eq. 5):

$150\,\mathrm{mm}$4

where $150\,\mathrm{mm}$5 is the measured resistance, $150\,\mathrm{mm}$6 and $150\,\mathrm{mm}$7 the calibration endpoints, $150\,\mathrm{mm}$8 the calibrated range (approx. $150\,\mathrm{mm}$9), and $\times$0 the fully-extended offset. Calibration achieves pin-specific transformation from resistance to displacement for accurate 3D mapping.

Measurement noise is modeled as $\times$1, with empirically observed values:

• Convex: $\times$2
• Concave: $\times$3
• Maximum: $\times$4
• Minimum: $\times$5

From the registered array data, a 3D point cloud is reconstructed (Eq. 6):

$\times$6

Here $\times$7 specifies the gripper’s base pose, and $\times$8 are grid indices. Noise-tolerant surface estimation applies voxel binning: for each cell $\times$9,

$165\,\mathrm{mm}$0

Advanced fusion—such as weighted least-squares or Gaussian-process regression—is also feasible.

4. Theoretical Gripping Analysis: Contact Mechanics and Friction

The gripping force on each pin is bounded by the frictional stick condition (Asbeck et al. model, Eq. 3):

$165\,\mathrm{mm}$1

with $165\,\mathrm{mm}$2 the tangential force, $165\,\mathrm{mm}$3 the friction coefficient, and $165\,\mathrm{mm}$4 the local asperity angle. Under ideal equal loading (total gripper pull force $165\,\mathrm{mm}$5 shared among $165\,\mathrm{mm}$6 pins per contact side):

$165\,\mathrm{mm}$7

The no-slip condition thus yields the maximum allowable holding force (Eq. 4):

$165\,\mathrm{mm}$8

Convex and concave terrains differ primarily in contact count $165\,\mathrm{mm}$9 and sign of $\times$0; more pronounced curvature (large $\times$1) results in increased $\times$2. The friction cone imposes a limiting angle $\times$3, ensuring the normal force vector remains within a critical angular sector relative to the local surface normal.

5. Empirical Validation and Performance Metrics

Validation encompasses grasping forces, terrain reconstruction accuracy, and theoretical-experimental consistency.

• Grasping Force vs. Terrain Slope: The gripper maintains stable forces $\times$4–$\times$5 for inclination angles $\times$6 (on sandpaper #40). At near-vertical slopes ($\times$7), holding force drops to $\times$8–$\times$9 with greater variability due to reduced contact ($70\,\mathrm{mm}$0) [Fig. 8]. Conventional "HubRobo" style grippers are less effective, especially on concave profiles.
• Shape Recognition Accuracy: When tested on emulated convex/concave arcs of $70\,\mathrm{mm}$1 height, per-pin standard deviations are $70\,\mathrm{mm}$2–$70\,\mathrm{mm}$3 (max $70\,\mathrm{mm}$4). The reconstructed curvature sign and general profile are systematically accurate [Fig. 10].
• 3D Mapping: For terrain of $70\,\mathrm{mm}$5 (including convex/concave sections and $70\,\mathrm{mm}$6 Y-slope), $70\,\mathrm{mm}$7 presses at $70\,\mathrm{mm}$8 produce approximately $70\,\mathrm{mm}$9 sampled points. Voxel-averaged height RMSE is $d_{\text{pin}}\approx2\,\mathrm{mm}$0, consistent with expected measurement noise. Outliers, largely attributable to single-pin error, are mitigated by bin-averaging [Fig. 12].
• Comparison with Theoretical Predictions: The relationship $d_{\text{pin}}\approx2\,\mathrm{mm}$1 is confirmed; higher curvature yields increased pull-off strength. The predicted total per-gripper holding force (based on $d_{\text{pin}}\approx2\,\mathrm{mm}$2, $d_{\text{pin}}\approx2\,\mathrm{mm}$3) aligns with observed $d_{\text{pin}}\approx2\,\mathrm{mm}$4–$d_{\text{pin}}\approx2\,\mathrm{mm}$5, accounting for real-world spine skipping and asperity fracture. Mapping error follows Gaussian additive noise with $d_{\text{pin}}\approx2\,\mathrm{mm}$6–$d_{\text{pin}}\approx2\,\mathrm{mm}$7.

6. Synthesis: Core Relations and Applications

The local displacement convex–concave structure operationalizes terrain-conforming grasp and tactile shape recognition as follows:

• Displacement: $d_{\text{pin}}\approx2\,\mathrm{mm}$8
• Elastic response: $d_{\text{pin}}\approx2\,\mathrm{mm}$9
• Frictional stick: $z=z(x)$0
• Gripper hold: $z=z(x)$1

A dense, independently calibrated array of sensorized pins enables both robust mechanical adaptation and point-wise 3D mapping with spatial resolution set by grid pitch and absolute accuracy $z=z(x)$2. This architecture addresses deficiencies of traditional fingered grippers on irregular terrain, allowing simultaneous grasping and real-time topographic measurement (Kato et al., 13 Jan 2026). The underlying structure generalizes to any scenario requiring conformal interface with unknown convex or concave surfaces using discrete, locally-sensed elastic elements.