Programmable Locking Cell in Robotics

• Programmable Locking Cell (PLC) is a modular, tendon-driven unit that switches between compliant and firm states through interlocking teeth for reconfigurable robot designs.
• By using tendon actuation and precise mechanical interlocking, PLCs achieve spatially programmable stiffness with up to a 9.5-fold tunability demonstrated in experiments.
• PLCs enable versatile applications such as variable-stiffness grippers and pipe-insertion robots, while highlighting challenges like discrete activation, backlash, and hysteresis.

Searching arXiv for the explicitly named "Programmable Locking Cells" paper and closely related usages of "PLC" to ground the article in current literature. A Programmable Locking Cell (PLC) is, in the robotics sense formalized by recent literature, a modular, tendon-driven robotic unit that can switch between compliant and firm mechanical states by mechanically engaging or disengaging interlocking teeth between adjacent structural elements. The concept is introduced as a structure-centric mechanism for achieving spatially programmable stiffness and morphological adaptability in modular robots operating in unstructured environments, where compliance is needed for adaptive grasping or confined-space navigation and rigidity is needed for shape holding, load bearing, or forceful manipulation (Zhou et al., 9 Sep 2025). The acronym is, however, polysemous across technical fields: in industrial control it conventionally denotes a Programmable Logic Controller, and in hardware security it is sometimes a plausible informal label for key-programmable locking primitives, although the cited logic-locking papers do not define a standardized architecture under the name “Programmable Locking Cell” (Maesschalck et al., 2022, Liu et al., 2024, Sweeney et al., 2021, Sweeney et al., 2020, Lopez et al., 3 Jan 2025).

1. Terminological scope and disciplinary usage

In robotics, the PLC is a unit cell architecture whose stiffness is modulated by mechanically interlocked joints actuated by cable tension. Each unit transitions between a compliant and a firm state, and multiple units can be assembled into reconfigurable robots with spatially programmable stiffness (Zhou et al., 9 Sep 2025).

Outside robotics, the same acronym is used differently. In industrial control systems, “PLC” refers to the Programmable Logic Controller targeted by tools for vulnerability discovery or automated code generation; those works concern controller software, memory structures, ladder logic, and Structured Text rather than mechanical locking units (Maesschalck et al., 2022, Liu et al., 2024). In logic-locking research for integrated circuits, several papers discuss programmable elements, key gates, MUX-based locking, LUT-based locking, and phase-programmable latches, but they explicitly do not present a named Programmable Locking Cell architecture in the sense of a standardized reusable PLC macro (Sweeney et al., 2021, Sweeney et al., 2020, Lopez et al., 3 Jan 2025).

This terminological divergence matters because the robotic PLC is a mechanical stiffness-programming primitive, whereas the industrial-control and hardware-security usages concern, respectively, controller computation and key-programmable circuit behavior. A plausible implication is that any encyclopedia treatment of PLC must distinguish acronym overlap from architectural equivalence.

2. Mechanical architecture of the robotic PLC

A robotic PLC unit is built from five independent structural components arranged around a central spine:

1. Upper spine
2. Tendon ring
3. Bearing
4. Locking ring
5. Lower spine (Zhou et al., 9 Sep 2025)

During assembly, the lower spine passes through the annular components and inserts into the upper spine, where a mechanical stage stops it at a prescribed location. The upper and lower spine are then bonded together in that local assembly step, while the annular parts remain free to move relative to the spine column with small axial and rotational clearance. The robot used in experiments was manufactured from ABS resin, the slip-fit interfaces were produced by SLA printing, the engagement surfaces were lightly surface-finished, and the resulting radial clearance was about 0.2 mm (Zhou et al., 9 Sep 2025).

The core locking function is generated by radial teeth on three elements: the lower spine of one unit, the locking ring, and the upper spine of the next unit. These teeth can mechanically interlock. The paper distinguishes two inter-unit states:

• Coupled: rotation of one unit forces synchronized rotation of the other.
• Decoupled: one unit can rotate relative to the other (Zhou et al., 9 Sep 2025).

The locking ring can rotate through $360^\circ$, but with $N$ teeth the stable locking positions become discrete at angular increments of $360/N$. In the reported prototype and planning procedure, the discrete angular set is

$[0,36,72,\dots,324]^\circ,$

which yields 10 discrete rotational states per unit (Zhou et al., 9 Sep 2025).

The reported geometric and material parameters for the inclined unit are:

• Young’s modulus $E$: $100\sim130$ MPa
• Spine outer diameter: $8\ \text{mm}$
• Spine inner diameter: $2\ \text{mm}$
• Spine length $l$: $30\ \text{mm}$
• Inclined angle $N$0: $N$1
• Origami skin outer diameter $N$2: $N$3
• Origami skin inner diameter $N$4: $N$5 (Zhou et al., 9 Sep 2025)

A through-hole in the spine allows a flexible shaft or other tools and cables to pass through the robot. Tendon actuation is implemented through a tendon ring; both tendon ends are tied to that ring, and tendon tension is applied by a linear actuator. A separate single flexible shaft driven by a rotary actuator passes through all units to provide rotational reconfiguration. Thus, the architecture separates stiffness actuation from motion actuation (Zhou et al., 9 Sep 2025).

An origami skin rigidly attached between adjacent tendon rings provides torsional resistance, allows bending in different directions for inclined units, and helps prevent buckling. This auxiliary structure is necessary because friction in the bearing and locking interfaces can otherwise rotate the tendon rings undesirably during actuation (Zhou et al., 9 Sep 2025).

3. Operating principle, state transition, and analytical model

The PLC operates by alternating between loosening stiffness and firmed stiffness. At zero or low tendon tension, adjacent spines are effectively decoupled, permitting relative rotation and compliant behavior. At sufficiently high tendon tension, the locking ring is pulled into engagement so that adjacent units behave as a single rigidly connected bent rod (Zhou et al., 9 Sep 2025).

The resulting programming sequence is mechanical rather than algorithmic in the software sense. A typical control sequence is:

1. keep all but one target joint locked,
2. release the target joint into loosening state,
3. rotate the distal body using the single flexible shaft,
4. stop at the nearest desired discrete tooth-aligned angle,
5. reapply tendon tension to lock,
6. move to the next joint and repeat (Zhou et al., 9 Sep 2025).

This makes the workspace inherently discrete. For a single inclined segment of length $N$6 and constant bending angle $N$7, the distal-end position of segment $N$8 in frame $N$9 is

$360/N$0

and the orientation is

$360/N$1

or explicitly,

$360/N$2

with $360/N$3, $360/N$4, $360/N$5, $360/N$6 (Zhou et al., 9 Sep 2025).

For $360/N$7 segments, the base-to-end homogeneous transform is

$360/N$8

with

$360/N$9

Because the joint angles are discrete, the authors use exhaustive state-space construction and a k-d tree / k-NN inverse-kinematics search to find reachable configurations close to desired points (Zhou et al., 9 Sep 2025).

In the firmed state, the PLC chain is treated as a unified elastic structure. For segment $[0,36,72,\dots,324]^\circ,$0,

$[0,36,72,\dots,324]^\circ,$1

with

$[0,36,72,\dots,324]^\circ,$2

Using Castigliano’s theorem, the end displacement is expressed as

$[0,36,72,\dots,324]^\circ,$3

The paper explicitly concludes that the firmed stiffness is anisotropic and configuration-dependent because it depends on all segment axial directions $[0,36,72,\dots,324]^\circ,$4 (Zhou et al., 9 Sep 2025).

The transition out of the firmed regime is modeled through a detachment threshold. External force $[0,36,72,\dots,324]^\circ,$5 applied at $[0,36,72,\dots,324]^\circ,$6 generates

$[0,36,72,\dots,324]^\circ,$7

while the resisting tendon torque is

$[0,36,72,\dots,324]^\circ,$8

At the home pose $[0,36,72,\dots,324]^\circ,$9, this becomes

$E$0

and detachment occurs when

$E$1

This model is used to distinguish the structurally load-bearing firmed state from the tendon-dominated post-detachment response (Zhou et al., 9 Sep 2025).

4. Experimental characterization and measured performance

The principal quantitative claim is a maximum stiffness variation ratio of up to 950% per unit, corresponding to a ratio of 9.5 (Zhou et al., 9 Sep 2025). In the normalized stiffness comparison reported in the appendix, the paper gives:

• Max original stiffness: $E$2
• Min original stiffness: $E$3
• Ratio: 9.5 (Zhou et al., 9 Sep 2025)

The text interprets this as “up to 950%,” which in context denotes a 9.5-fold tunability ratio. This suggests a high contrast between compliant and firmed operation, even though the mechanism is fundamentally discrete in tooth engagement.

Single-segment characterization used force–deformation measurements. The reported values include:

• Single-segment firmed stiffness prediction: $E$4 to $E$5
• Single-segment measured firmed stiffness: about $E$6
• Tendon tensions used in test: $E$7, $E$8, $E$9
• Observed loosening-threshold force intervals:
  • $100\sim130$0 tendon tension: $100\sim130$1
  • $100\sim130$2: $100\sim130$3
  • $100\sim130$4: $100\sim130$5
• Theoretical threshold relation:

$100\sim130$6

• Tendon connection radius on lower ring: $100\sim130$7
• Pushing point location for threshold estimation: approximately $100\sim130$8 above the fulcrum (Zhou et al., 9 Sep 2025)

The measured force–displacement curves show an initial firmed-stiffness slope, a separation point, a lower-slope loosening regime dominated by tendon stretch, and hysteresis attributed to friction, material deformation, and elastic/inelastic tendon interactions (Zhou et al., 9 Sep 2025).

Multi-segment tests examined stiffness in eight directions in a plane perpendicular to a segment axis, at

$100\sim130$9

The purpose was to verify that end stiffness decreases as chain length grows and remains anisotropic and configuration-dependent. The paper states that theory somewhat overestimates measured stiffness because of backlash and tolerance accumulation (Zhou et al., 9 Sep 2025).

Torsional testing identified another nonideality: the origami skin starts slight buckling at approximately $8\ \text{mm}$0 torque (Zhou et al., 9 Sep 2025). The paper also uses $8\ \text{mm}$1 as the external force in anisotropy simulation (Zhou et al., 9 Sep 2025).

Several quantities are explicitly not reported and therefore remain outside the documented characterization: unit mass, total robot weight, response or switching time, durability or cycle-life, repeatability statistics, exact tooth-interface locking force, exact maximum bending moment, tendon material specification, actuator models, control bandwidth, and quantitative in-hand manipulation accuracy (Zhou et al., 9 Sep 2025).

5. System-level embodiments: gripper and confined-space robot

The first functional prototype is a variable-stiffness gripper with 2 fingers and 3 PLC units per finger (Zhou et al., 9 Sep 2025). In low-stiffness mode, the fingers conform to irregular or deformable objects, including sponges and toys. In high-stiffness mode, the same gripper securely holds heavier objects, explicitly including a 2 kg water bottle and a 1 kg weight (Zhou et al., 9 Sep 2025).

The paper also describes a simple form of in-hand manipulation enabled by selective release of only the final PLC unit in a finger. This allows either larger-scale whole-finger rotation or more local fingertip rotation, although no quantitative manipulation metrics are reported (Zhou et al., 9 Sep 2025).

The second prototype is a 16-unit pipe-traversing or pipe-insertion robot (Zhou et al., 9 Sep 2025). Its architecture is divided into:

• rear 13 segments with unified stiffness control
• front 3 segments with independent stiffness tuning (Zhou et al., 9 Sep 2025)

This organization supports a deployment strategy in which the long rear body remains compliant enough to adapt to confined geometry, while the front section is selectively stiffened for distal operations such as screw tightening. The demonstrated outcomes are successful pipe insertion with compliant body behavior and successful distal screwing after local stiffening (Zhou et al., 9 Sep 2025).

These prototypes illustrate the architectural claim that PLCs enable mechanical modularity, stiffness modularity, morphological modularity, and functional modularity through serial composition and selective locking. The paper also states that reconfigurations include C-shapes, S-shapes, adaptive wrapping morphologies, and stiffness distributions such as a compliant proximal body with a stiff tip (Zhou et al., 9 Sep 2025). A plausible implication is that the PLC is intended less as a single isolated joint than as a repeatable structural primitive for continuum-like robotic bodies.

6. Tradeoffs, limitations, and relation to adjacent “PLC” concepts

The paper positions the robotic PLC against antagonistic actuation, phase-change materials, jamming, and prior structure-centric locking mechanisms. The stated advantages are modularity, discrete stiffness programmability, low energy consumption in the locked state, high stiffness ratio, and robustness under high load; the main tradeoffs are that stiffness control is discrete rather than truly continuous, and that current implementations still exhibit backlash, frictional nonidealities, and coupling introduced by continuous tendon routing (Zhou et al., 9 Sep 2025).

Several limitations are explicit. The present implementation uses a continuous tendon routed across multiple units, which reduces segment independence and limits modularity. The robot exhibits hysteresis, friction between tendon ring, bearing, and locking ring, and discrepancy between theory and multi-segment measurements due to accumulated backlash. The workspace and locking states are inherently discrete because they depend on tooth engagement. Dynamic performance was not fully characterized, including switching latency, repeatability under cyclic loading, and closed-loop dynamic response (Zhou et al., 9 Sep 2025).

The broader term “PLC” remains a source of confusion. In hardware security, logic-locking papers discuss key-programmable structures that are nearest in spirit to a “programmable locking primitive,” including XOR/XNOR key gates, MUX locking, LUT-based locking, densely interconnected configurable logic/routing blocks, and phase-programmable latches, but those papers emphasize that no explicit standard-cell or named Programmable Locking Cell is introduced (Sweeney et al., 2021, Sweeney et al., 2020, Lopez et al., 3 Jan 2025). In industrial control, the same acronym refers to Programmable Logic Controllers, as in PLC-VBS for vulnerability discovery and Agents4PLC for automated Structured Text generation and verification; these are semantically unrelated to the robotic unit cell despite the shared abbreviation (Maesschalck et al., 2022, Liu et al., 2024).

Accordingly, the most precise encyclopedic definition is domain-specific. In robotics, a Programmable Locking Cell is a modular tendon-driven unit whose interlocking teeth enable discrete stiffness modulation and morphological adaptability under selective tendon actuation (Zhou et al., 9 Sep 2025). In adjacent fields, “PLC” either denotes a controller platform or serves, at most, as an informal interpretive label for other programmable structures rather than a standardized architecture (Maesschalck et al., 2022, Liu et al., 2024, Sweeney et al., 2021, Sweeney et al., 2020, Lopez et al., 3 Jan 2025).