Force-moment mechanics of wiggling in connector insertion

Abstract

Humans instinctively use both pushing and wiggling motions when inserting objects into sockets, connectors, or mechanical interfaces that resist fitting, often without understanding the rationale behind these actions. Despite their apparent simplicity, such tasks involve complex mechanical interactions that are challenging to describe mathematically. In this paper, we present a simplified two-leg mathematical model that clarifies the fundamental principles governing the combined pushing-wiggling during assembly. The model integrates forces, moments, frictional reactions, and other uncertain effects into several effective factors representing overall mechanical behavior, classifying the relationship between these factors and connector movements into nine cases and demonstrating that the coordinated application of force and moment facilitates insertion. The experimental validation confirms the effectiveness of the model and highlights the role of wiggling in overcoming insertion resistance. Moreover, the model successfully generalizes across various connector types, pin counts, and configurations, offering insights into the broader mechanics of human-inspired insertion strategies.

The authors explain why people wiggle connectors during insertion. A simple two-leg mechanical model shows combining vertical force with small rotational moments overcomes friction, reducing effort, jamming, and enabling efficient assembly.

Introduction

Assembling an electrical connector into its mating socket, inserting a key into a lock, or plugging a USB into a port are common daily activities that may seem deceptively simple. However, when an object does not fit smoothly into its intended receptacle, an instinctive response often occurs: we subconsciously wiggle the object while applying pressing force. This combined action, typically performed intuitively without deliberate awareness or deep understanding of the mechanics involved, can facilitate smoother insertion with less force compared to attempts without wiggling. Wiggling can mitigate issues such as jamming or wedging in specific scenarios. Despite its ubiquity, the fundamental principles underlying the effectiveness of wiggling as a strategy to overcome insertion or removal resistance have not been extensively investigated.

The act of wiggling during insertion involves more than simply applying additional force or rocking a connector to find a path of least resistance. It entails complex interactions among applied forces, moments, material properties, and frictional reactions at contact points. These interactions are further complicated by factors including object shapes, surface roughness, misalignments, and more. Understanding these nuances not only provides insights into a universal yet elusive phenomenon but also facilitates the design of more efficient assembly processes and the development of machines capable of replicating human dexterity in manipulation tasks.

Various classical friction models have been employed to characterize static, dynamic, and sliding friction between contacting bodies. Building upon these models, conventional assembly studies have primarily focused on insertion mechanics from a force-based perspective, emphasizing input force, friction, and geometry^1–3. Assuming that assembly is conducted at sufficiently low speeds, the process has been modeled as a quasi-static interaction, analyzing rigid body contact forces and geometry in a two-dimensional plane, along with a jamming diagram⁴. Expanding on the quasi-static contact model, researchers have also examined contact forces and states for various configurations, including dual-peg, triple-peg, and elastic structures^5–8. Regarding jamming phenomena, multiple studies have both predicted and mitigated such contact conditions, thereby supporting reliable and successful peg-in-hole insertions^9–11. Beyond focusing solely on contact forces, researchers have analyzed moment data to develop force/torque maps, estimate contact position errors, and devise insertion strategies to compensate for these misalignments^12–14. Although moments have been actively monitored and examined during the assembly process, these studies have not clarified the fundamental principles governing their roles and effects.

Numerous investigations have examined the impact of vibration on frictional behavior, proposing strategies to reduce resistance and improve assembly efficiency^15–22. Additionally, other researchers have developed vibration techniques for socket mating that capitalize on these effects^23–27. However, vibration approaches generally depend on translational motion, either along the insertion axis or perpendicular to it, to facilitate assembly in experimental setups. Most analyses are retrospective studies based on experimental results rather than model-based methodologies. Furthermore, although these experimental approaches are effective and demonstrate reductions in insertion resistances, they generally treat vibration as a practical tool to overcome high resistance, without providing explicit friction models to clarify how oscillatory motion assists connector insertion. In particular, they do not address the rotational wiggling strategies observed in human behavior and their effects.

More recently, robotic applications for precise positioning and insertion of objects have been explored, including pick-and-place tasks, peg-in-hole insertion, connector alignment, and wire harnessing^28,29. One of the most common approaches for implementing these robotic applications involves utilizing a vision system to identify stable grasping positions and determine optimal initial contact points^30–34. While vision-based systems have enabled various assembly strategies that leverage object geometry and spatial modeling, they often overlook the essential mechanics of socket insertion, such as contact force analysis and underlying physical principles. Peg-in-hole tasks are the quintessential robotic applications, and numerous model-based methods have been developed to reduce insertion failures by enhancing alignment accuracy. In particular, several strategies have integrated active compliant control with force feedback from force/torque (FT) sensors. Among these, many researchers have employed impedance control, utilizing reaction force data from FT sensors to successfully execute peg-in-hole tasks by regulating the dynamic interplay between force and motion, thereby allowing compliant interactions with the environment during insertion^35,36.

A guidance algorithm combining impedance control with a vision system and geometric information has been applied to handle various complex-shaped objects³⁷, as well as to manipulate and assemble flexible rubber objects with unknown dynamics through force control–based methods³⁸. A tilt strategy incorporating impedance control was developed for electric connector assembly, leveraging force data from FT sensors alongside image-based visual servoing³⁹. Additionally, a multi-fingered robotic hand equipped with a sensor-rich, force-based control strategy has been utilized for peg-in-hole tasks, providing a benchmark against conventional force-controlled gripper systems⁴⁰. Rather than achieving compliance through control methods, numerous studies have explored assembly strategies using passive compliance mechanisms, such as remote center compliance (RCC) devices^41–43 and other supplementary systems^44–46. Recently, learning-based approaches have integrated the robot’s dynamic model with various sensor inputs, particularly reaction forces measured by force–torque (FT) sensors, to adaptively handle diverse insertion scenarios^47–50. Unlike pre-programmed motion sequences, these data-driven methods aim to generalize across varying contact conditions, enabling more flexible and robust socket insertion. Although assembly tasks have been a central focus in robotics research for some time, efforts have primarily targeted strategy development to simplify and streamline these operations without elucidating the mechanics of socket mating, especially the human rotational wiggling behavior.

Overall, although prior studies enhance the understanding of friction, contact mechanics, and the fundamental principles of insertion, they do not sufficiently explain human activities during assembly. Specifically, they have not attempted to analyze how a slight rotational action, combined with a pushing force, can facilitate insertion under tight-fit conditions. This intuitive wiggling behavior is ubiquitous in everyday connector use, and its widespread effectiveness strongly suggests that rotational action combined with vertical pressing force plays an important role in connector assembly. This motivates a closer mechanical analysis of the phenomenon and the development of engineering insights that can be used to automate precision connector assemblies in manufacturing processes.

In this paper, we present a comprehensive study on the role of a wiggling moment superimposed on pushing force during socket assembly and disassembly, highlighting the fundamental principles governing these simultaneous actions. Drawing on systematic observations of human pushing-wiggling behavior during insertion and removal, we identified four recurring phases. Building on these observations, we developed and validated a two-leg model that provides a mechanistic explanation of this behavior, although minor discrepancies may arise in practical applications. The model was first implemented on a simple 2-pin connector and then extended to various types, including 3-pin, 25-pin connectors, and a power plug, all of which exhibited behaviors that were consistently explained by the model and were successfully assembled and disassembled. These results demonstrate that the model is broadly applicable regardless of connector types, pin counts, or object configurations. Furthermore, its application to robotic automated connector assembly and disassembly enhanced efficiency while reducing potential damage, highlighting its value for safer and more reliable automation.

Results

Instinctive human behavior in connector assembly

Most individuals experience the frustration of connectors, plugs, or sockets failing to engage smoothly, even with considerable force. In such situations, individuals instinctively combine a downward push with a small rotational action (wiggling) of the connector—acquired through repeated experience rather than formal instruction, even though they typically do not understand the mechanical rationale for this action (Fig. 1). Our observations of connector assembly and disassembly under significant resistance indicate that this lateral wiggling around the connector’s center, together with the downward force, causes the connector to alternately advance or retreat left and right relative to the mating receptacle, thereby greatly facilitating insertion or removal. More detailed analysis of human insertion and extraction is included in Supplementary Note 1, Supplementary Movie 1.

Fig. 1. Instinctive human behavior during socket assembly.

Socket assembly is a common everyday task, such as plugging in electronic devices, connecting electrical circuits, or assembling furniture. Observations show that people instinctively apply a pushing force while simultaneously wiggling the object laterally during insertion.

Four distinct phases of the wiggling behavior were identified during the human assembly of a tightly resisting connector, as shown in Fig. 2b. The first phase, vertical force only (VFO), occurs when only vertical force is applied during the initial stage of connector assembly. If the force is insufficient to overcome the connector’s resistance, the connector remains stationary or barely advances. In the second phase, moment-assisted insertion 1 (MAI-1), an additional rotational moment is applied through the wiggling action while maintaining the vertical force, rather than merely increasing the pressing force. This causes one side of the connector to begin advancing, while the other side shows minimal movement. If the connector rotates to the mating tolerance limit and the applied moment becomes excessive due to increased rotational resistance, the third phase, over-rotation locking (ORL), occurs, immobilizing the connector and preventing any further rotational or downward movements. During moment-assisted insertion 2 (MAI-2), the applied rotational moment rapidly decreases and then reverses direction, causing the connector component to return to its vertical orientation. At this stage, the side advanced in MAI-1 remains stationary, whereas the opposite side begins a downward movement until the connector fully reverts to its initial vertical orientation (VFO). By sequentially progressing through the phases, MAI-1 → ORL → MAI-2, the left and right sides are alternately inserted in a coordinated manner. A continuous progression of these alternating forward motions leads to the successful completion of the connector assembly.

Fig. 2. Phase observations of connector insertion during human wiggling.

Detailed observation of the connector insertion process using combined pushing and wiggling actions. a Common two-pin connector used for observation. b Connector movements observed when a user simultaneously applies pushing and wiggling actions during insertion. During insertion, a moment is applied to the upper part of the connector in the direction of rotation. One wiggling cycle can be divided into four distinct phases (VFO, MAI-1, ORL, MAI-2); sequential alternation through these phases results in complete connector assembly.

Building on observations of human behavior during connector assembly, this study explores the mechanical principles underlying the moment effect generated by wiggling, highlighting its benefits and role in the assembly process. A hypothetical two-leg model for connector assembly is presented to facilitate the mathematical analysis and is applicable to various connector types regardless of pin shape, number, or configuration due to their common physical properties. Finally, the study proposes an insertion strategy that improves assembly efficiency and stability by utilizing the interplay between applied vertical forces and rotational moments through wiggling.

Two-leg model

In any general assembly task, there are complex physical interactions between the mating parts, including exchanges of frictional forces, normal reactions, surface irregularity resistances, misalignment-induced torques, and bending forces on the housing and pins. These interactive forces are influenced by multiple factors, including contact points and areas, surface characteristics, flexibility, and tolerances. Due to their complexity, defining and mathematically modeling them in their original form is nearly impossible. Consequently, we developed a two-leg connector model that idealizes a general connector, as illustrated in Fig. 3a. This model consolidates numerous, often unmeasurable variables into a few representative forces and moments. While preserving the principal characteristics of complexly shaped connectors, the two-leg model clarifies how the wiggling moment imposed on the pressing force affects the connector assembly. In subsequent analyses using the idealized two-leg model, we assume that socket assembly is a quasi-static task, focusing on the steady-state relationships between the interactive forces. This assumption reflects the slow, precision assembly conditions under which inertial and other dynamic effects are small compared with contact and friction forces. Insertion can be approximated as a sequence of quasi-static, stick-slip equilibria at the onset of motion. In typical human assembly tasks and high-speed automated assembly, a more detailed dynamic model may be required to capture all inertial and velocity-dependent effects, and our quasi-static analysis may be less accurate quantitatively in this regime; however, it reliably captures the overall sequence of connector motions and the underlying mechanism by which wiggling facilitates insertion and extraction.

Fig. 3. Free-body diagram of two-leg connector model and nine cases based on the movement of the legs.

a Free-body diagram of the two-leg model. b Nine cases: the horizontal axis is the vertical force $F_h$, the vertical axis is the moment-induced force $Q_h$, and the $F_h-Q_h$ plane is divided into nine regions by the four boundary lines connecting $\left(-2f_s^{\max },0\right)$ to $\left(\mathrm{0,\; 2}{f}_s^{\max }\right)$, $\left(\mathrm{0,\; 2}{f}_s^{\max }\right)$ to $\left(2f_s^{\max },0\right)$, $\left(2f_s^{\max },0\right)$ to $\left(0,-2f_s^{\max }\right)$, and $\left(0,-2f_s^{\max }\right)$ to $\left(-2f_s^{\max },0\right)$. c Movements of the legs induced by the applied moment in cases 2–7.

Initially, we define the variables associated with the proposed two-leg model in Fig. 3a. The force $F_h$ denotes the vertical force applied by the human, and $M_h$ represents the rotational moment generated by the human’s wiggling action. Reaction forces $f_l$ and $f_r$ are applied to the left and right legs, respectively, countering the external force/moment. The forces $f_l$ and $f_r$ can be regarded as the driving forces of the legs for movement. By achieving force equilibrium along the vertical axis, we obtain:

$$F_h=f_l+f_r$$ 1

These reaction forces equal the vertical resistances in the left and right legs, opposing the vertical force $F_h$ in a static situation. They are predominantly generated by the friction between male pins and female sockets, and possibly between housings. The wiggling motion begins when a human applies an external moment $M_h$ to the upper connector component, triggering a reactive moment to counteract it. This reactive moment arises from both vertical reactions $f_l$ and $f_r$ and a lateral reaction force $F_y$, likely due to bending. Assuming a small rotation angle, θ, limited to a few degrees—where $\cos \mathrm{\theta }\approx 1$ and $\sin \mathrm{\theta }\approx 0$—the moment equilibrium about the center of rotation along the x-axis can be expressed as follows.

$$M_h-r\cdot f_l+r\cdot f_r-d\cdot F_y=0$$ 2

where r denotes half the distance between the two legs, and d indicates the distance from the center of rotation to the point where the lateral force $F_y$ acts. By nature, $F_y$ increases resistance as the connector rotates toward the limit of one side.

Solving Eqs. (1, 2) simultaneously yields the vertical reaction force for each leg as

$$f_l=\frac{F_h}{2}+\frac{M_h-d\cdot F_y}{2r},f_r=\frac{F_h}{2}-\frac{M_h-d\cdot F_y}{2r}$$ 3

or simply

$$f_l=\frac{F_h+Q_h}{2}and{f}_r=\frac{F_h-Q_h}{2},with{Q}_h=\frac{M_h-d\cdot F_y}{r}$$ 4

where $Q_h$ represents the linear force induced by moments such as $M_h$ and $d\cdot F_y$. Given that r is only a few millimeters for typical connectors, a human’s rotational effort $M_h$ is significantly amplified and transmitted into $Q_h$. When a vertical pressing force $F_h$ is applied during the connector assembly process, it is uniformly distributed to $f_l$ and $f_r$, establishing a forward bias. The moment-induced force, $Q_h$, acts anti-symmetrically. Specifically, for the left leg, the moment-induced force $Q_h/2$ aids in overcoming the static friction and promotes forward movement. Conversely, it hinders the forward motion or, in severe cases, reverses the movement of the right leg by opposing the vertical force.

For systematic analysis, consider each leg’s general state—stationary, moving forward, or moving backwards—based on the relative magnitudes of $f_l$ and $f_r$ in relation to the maximum static friction, $f_s^{max}$.

1. Stationary: $-f_s^{max}<f_r,f_l<f_s^{max}$

2. Forward: $f_r,f_l<-f_s^{max}$

3. Backward: $f_r,f_l>f_s^{max}$

When both legs are considered simultaneously, the connector’s state can be categorized into nine cases (three for the left leg × three for the right leg), as illustrated in Fig. 3b. These cases are listed as {$\left(\text{left},\text{right}\right)\text{∣}\left(\text{F},\text{F}\right),\left(\text{F},\text{S}\right),\left(\text{S},\text{F}\right),\left(\text{F},\text{B}\right),\left(\text{B},\text{F}\right),\left(\text{B},\text{S}\right),\left(\text{S},\text{B}\right),\left(\text{B},\text{B}\right),\left(\text{S},\text{S}\right)$}, where F, S, and B represent forward, stationary, and backward states, respectively.

Furthermore, it is advantageous to redefine these nine cases in terms of $F_h$ and $Q_h$, based on Eq. (4), as depicted in Fig. 3b. Case 1 (left = F, right = F) satisfies $F_h+Q_h>2f_s^{max}$ and $F_h-Q_h>2f_s^{max}$; this scenario occurs when substantial positive vertical pressure is applied, but the rotational effort, regardless of its direction, remains relatively insignificant. Case 2 (left = F, right = S) occurs when $F_h+Q_h>2f_s^{max}$ and $\mid F_h-Q_h\mid \le 2f_s^{max}$, indicating that both vertical pressure and rotational effort are positive with moderate magnitudes. Case 3 (left = S, right = F) arises when $\mid F_h+Q_h\mid \le 2f_s^{max}$ and $F_h-Q_h>2f_s^{max}$, characterized by positive vertical pressure and negative rotational effort, both of moderate magnitude. These Cases 2 and 3 correspond to the MAI-1 and MAI-2 patterns, respectively, observed during the typical human connector insertion task detailed in the preceding section. Case 4 (left = F, right = B) pertains to $F_h+Q_h>2f_s^{max}$ and $F_h-Q_h<-2f_s^{max}$, distinguished by a substantial positive rotational effort coupled with negligible vertical pressure, irrespective of its sign.

Reversing the rotational effort direction from Case 4 defines Case 5 (left = B, right = F). Similarly, reversing the vertical pressure direction from Cases 2 and 3 produces their symmetric counterparts, Case 6 (left = B, right = S) and Case 7 (left = S, right = B), respectively. A significant reverse vertical pressure coupled with minimal rotational effort leads to Case 8 (left = B, right = B). Conversely, when both the vertical pressure and rotational effort are insufficient to overcome the maximum static friction, or if the connector jams as rotation approaches its mechanical tolerance limit, the entire connector remains stationary, resulting in Case 9 (left = S, right = S). This corresponds to the VFO or ORL phases, respectively, as observed during manual connector insertion by a human operator. The conditions and physical implications of all cases are readily understood through the symmetry of the left and right legs and the positive and negative directions of the vertical pressure and rotational effort, as illustrated in Fig. 3b, c.

In the context of human-driven tight connector assembly, Cases 2 and 3 of the two-leg model effectively elucidate Moment-Assisted Insertion, characterized by alternating side-to-side forward movements. By inverting the moment-induced force, a repeating sequence of Cases 2 and 3 is achieved, emulating the wiggling action observed during actual connector assembly. Similarly, a cyclical pattern of Cases 6 and 7, produced by systematically alternating the rotational moment with a vertical pull, parallels the wiggling action used in connector disassembly. Additionally, during real assembly, when the rotational moment is significant, and the vertical pressing force is minimal, simultaneous opposite up-down movements on both sides of the connector may occur, akin to Cases 4 and 5.

The aforementioned evidence and theoretical analysis indicate that the simplified two-leg model and its nine scenarios capture the characteristics of human-mediated connector assembly and reproduce the observed behaviors under applied forces and moments. Minor discrepancies may arise owing to variations in contact conditions, unobserved microscale force interactions, and potential dynamic effects. We employed a Coulomb-type simplified friction model because our analysis is based on a quasi-static formulation of the conditions under which the connector begins to move. However, in situations where the connector undergoes continuous motion, a more detailed friction model would be required to account for the dynamic effects, including Stribeck behavior and other velocity-dependent friction effects. The proposed mechanism and analyses are sufficiently robust to explain the core principles of connector insertion.

Experimentation: two-pin connector assembly

The experiment aimed to apply the pushing-and-wiggling insertion strategy utilized by humans for mating tight connectors within a precisely controlled environment and to validate whether such behavior could be explained by the proposed two-leg force/moment model. Using the device illustrated in Fig. 4, we conducted an experiment to evaluate the proposed two-leg model and its connector movement scenarios.

Fig. 4. Design of the experimental device.

a Front view of the device. A vertical force is applied either by grasping the handle or by placing weight disks on the top seat. Movement of the handle is constrained by a linear guide that permits only vertical motion, eliminating any off-axis forces. b Side view of the device. Laser sensors fixed in front of and behind the lower connector part measure the displacements of the left and right sockets, respectively.

The procedure started by applying a relatively low vertical force $F_h$ of 25 N to the handle during the initial phase. Considering the manufacturer’s recommended insertion force of 50 N, the applied vertical force was insufficient to initiate insertion, thereby preventing further socket advancement. Subsequently, a motor-controlled cyclic angular movement was applied to the upper mating component (female connector). This combined action caused the upper socket to move downward with a wiggling motion. Reapplying the push alongside the cyclic rotational movement ultimately resulted in complete socket insertion.

Figure 5 presents graphs of key physical quantities as the insertion sequence progressed. Figure 5a displays the applied vertical force $F_h$ and induced lateral force $F_y$ measured over time by the FT sensor. At $t=5\mathrm{s}$, a total mass of 2 kg was placed on the top seat of the handle to impose a constant vertical force. Subsequently, at $t=10\mathrm{s}$, a sinusoidal rotational motion with an amplitude of ±1.5° and a period of 6 s was applied to the upper connector, as depicted in Fig. 5b. This sinusoidal rotation with well-defined amplitude and period was used as a simple, repeatable way to emulate human wiggling. The amplitude was selected based on the tolerance of the two mating components, such that the connector undergoes a small angular motion within the clearance without excessive bending or repeated side contact. The period was set relatively slowly to accurately measure the related physical quantities under quasi-static conditions while keeping the inertial and dynamic effects small; shorter periods introduce noticeable dynamic effects, whereas longer periods yield similar force profiles but impractically long trials. The net moment $M_h$ delivered by the motor is depicted in Fig. 5c, representing the pure interaction moment, calculated by subtracting the no-load torque ${\tau }_{no-load}$ from the commanded motor torque ${\tau }_{cmd}$. The no-load torque ${\tau }_{no-load}$ was determined by performing the same sinusoidal motion without a load attachment. Detailed information on the calculation of this net moment and its characteristics is provided in the Supplementary note. As shown, $M_h$, the interaction moment appeared as a distorted sine wave and increased as mating progressed. The moment-induced force $Q_h$, obtained by converting $M_h$ using Eq. (4), is presented in Fig. 5d. The induced force profile $Q_h$ closely resembled that of $M_h$; however, its magnitude exceeded 100 N due to the leverage effect. Although $Q_h$ may have been affected by uncertainties from lateral forces or structural effects such as bending and coupling, it was primarily governed by the interaction moment $M_h$.

Fig. 5. Two-pin connector insertion results.

a Applied vertical force $F_h$ and lateral force $F_y$ arising from contact between two mating parts. b Desired and measured rotation angle of the upper connector part. c Net moment $M_h$ applied to the upper connector part. d Moment-induced force $Q_h$. e Socket displacements $L_l$, $L_r$ and wiggling angle θ. f Reaction forces $f_l$, $f_r$ acting on the left and right sides and static friction force $\mid f_s^{\max }\mid$. g Enlarged views of the early and late intervals of the connector insertion sequence.

The alternating left and right forward movements of the upper socket are depicted in Fig. 5e, where the displacements on the left and right sides are represented by $L_l$ and $L_r$, respectively. To elucidate the relationship between forward movement and cyclical rotation, the motor angle θ was superimposed. It was observed that the left and right sides of the upper connector descended sequentially in synchronization with the cyclic rotation. As expected, the left side of the socket moved downward when turned to the left, while the right side remained nearly stationary, and vice versa. The reaction force on the moving side was sufficiently large to exceed the maximum static friction force $f_s^{max}$ (see Fig. 5f), whereas the reaction force on the stationary side did not exceed the maximum static friction force $f_s^{max}$. As cyclical rotation continued, $f_l$ and $f_r$ alternated in direction with respect to the mean preload $F_h/2$.

Figure 5g presents the initial and final stages of the connector insertion process with magnified views. Before the onset of rotational movement, the left and right displacements $L_l$ and $L_r$ remained unchanged, and the corresponding reaction forces $f_l$ and $f_r$ stayed within the static friction limit $\mid f_s^{max}\mid$. This condition corresponds to Case 9 (S, S) in Fig. 3b, representing a stationary state where the applied force is insufficient to overcome static friction. Subsequently, upon initiating counterclockwise (CCW) rotation, opposite reactions developed through $M_h$, which caused the left side to advance by approximately 0.2 mm while the right side exhibited no significant movement. This behavior aligns with Case 2 (F, S), as indicated by the light orange shading. Following the complete CCW rotation, clockwise (CW) rotation commenced at approximately $t=11.5\mathrm{s}$. At this juncture, the two reaction forces intersected at the mean bias, reversing the states of the left and right sides. Consequently, due to the reversed moment, the right side advanced by approximately 0.4 mm, while the left side exhibited no significant movement. This behavior was the direct opposite of the preceding CCW rotation and corresponded to Case 3 (S, F), as indicated by the light blue shading. The photographic configurations of the upper and lower connector parts centrally display the behaviors of the left and right sides as the rotation direction changes.

As insertion advanced to later stages, the previously observed alternating pattern of sequential engagement persisted. However, some differences became inevitable at greater insertion depths due to a tighter fit between the connector parts. The reduced mechanical tolerance likely necessitated a higher applied moment to sustain rotation (refer to the latter portion of Fig. 5c, d). Consequently, reaction forces on both sides markedly increased compared to early stages, often surpassing the friction threshold, even opposing the insertion direction. Specifically, during the counterclockwise (CCW) rotation phase, the left side continued to advance while the right side occasionally retracted, corresponding to Case 4 (F, B) (highlighted in dark orange). Conversely, during the clockwise (CW) rotation phase, the right side advanced while the left side sometimes moved backwards, corresponding to Case 5 (B, F) (highlighted in dark blue). Despite temporary local retractions, the consistent pressing force $F_h/2$ continued to bias both $f_l$ and $f_r$ and successfully completed the insertion.

The experimental results aligned with observed human behavior during pressing accompanied by cyclic rotation, as used in the assembly and disassembly of tightly fitted components. The alternating behavior of part insertion and its underlying mechanism were effectively elucidated using the proposed two-leg mechanical and mathematical framework. The findings suggest that the rotational moment $M_h$ plays an important role in facilitating connector insertion, particularly when axial force alone is no longer sufficient to drive further insertion. Additionally, the symmetric and alternating forces generated by $M_h$ aid in overcoming static friction.

In addition to the LCB30 2-pin connector presented in this manuscript, additional insertion/extraction experiments with multiple connector types, repeated-trial results under identical conditions, and the corresponding quantitative summaries and statistical analyses are provided in the Supplementary Information. Across these additional evaluations, the measured force–moment–displacement signatures consistently exhibited the same overall pattern, indicating that the proposed mechanism and two-leg model are applicable across connector types and form factors.

Discussion

This study investigated the mechanical role of rotational motion, typically observed as the natural human behavior of wiggling during socket insertion tasks. Building on this behavior, we developed a streamlined two-leg model that clarifies how vertical force and user-generated moment interact with connector components and how their combination provides mechanical advantages during insertion. The model was analytically derived and experimentally validated using a custom device. Results confirmed that a moderate rotational moment, applied within mechanical tolerances, effectively enables sequential insertion by generating alternating moment-induced forces that assist in overcoming static friction.

By experimentally demonstrating that an optimal combination of human-applied force and moment induces the proposed left-right sequential insertion, we further elucidate the physical roles and effects of $F_h$ and $M_h$. For instance, when a relatively small $F_h$ is applied, the bias term $F_h/2$ acting on each socket becomes negligible, causing the mean values of $f_l$ and $f_r$ to approach zero. Consequently, the forward-driving force on each socket decreases while the tendency for the sockets to retract increases. This leads to reduced forward movement of the sockets and an increase in backward withdrawal, ultimately prolonging the overall insertion time. Conversely, applying a relatively small $M_h$ diminishes the magnitudes of both $f_l$ and $f_r$, allowing reaction forces to remain within static friction limits for an extended period. In this scenario, neither forward nor backward motion easily overcomes friction, causing the connector to advance slowly and prolonging the insertion process. Therefore, when force and moment are constrained or when inserting particularly tight-fitting sockets, the optimal insertion strategy involves selecting appropriate $F_h$ and $M_h$ values such that, in the forward direction, the net reaction force on a driven socket exceeds its static friction threshold, while the force on the opposing socket stays within the frictional limit. Under these conditions, the connector advances smoothly with minimal backward slip, completing the mating process in the shortest possible time and with minimal effort.

The selection of $M_h$ should be constrained by the allowable angular clearance and mechanical tolerance of the mating connector pair. If the applied moment or the resulting angular excursion is too large, the lateral loads can bend pins, distort housings, or place excessive stress on the attached cables. In this study, this range was enforced by monitoring the lateral force measured using an FT sensor and requiring that this lateral load, which increases when side contact intensifies near the clearance limit, remain below a predefined threshold. This threshold was set based on preliminary non-damaging trials and visual inspection. The two-leg model indicates that $M_h$ is mechanically converted into differential leg reaction forces; therefore, a relatively small rotational moment is sufficient to trigger friction-threshold crossings and produce the desired sequential engagement. Accordingly, we limited the applied rotation and moment to this tolerance-bounded window, and no visible damage or permanent deformation of the connectors or housings was evident over repeated trials.

The two-leg model simplifies the mechanical interaction between connector pins and sockets by representing the integrated reaction forces acting on the left and right sides of the connector body as two distinct legs. Each leg aggregates the combined resistance from the pins located on that side, thereby reducing the complex multi-pin contact problem to two representative forces. Since these leg forces are treated as lumped reactions, the model is generalizable regardless of the number of pins per leg, their thickness or length, or the detailed geometry of the connector. Furthermore, a single rotational moment applied by a human or an actuator can be equivalently expressed as a pair of opposite linear forces acting on the left and right legs. This transformation allows the model to describe the resulting mechanics independently of connector geometry, making it applicable to a wide range of connector assembly and disassembly tasks. The experimental validations of the model, including its application to the insertion and removal of various connectors with multiple pins or different shapes, are detailed in the Supplementary note and Supplementary Movie.

Adding a rotational moment to the insertion process may appear more efficient, as most users notice a reduction in the vertical force, while the specific contribution of the moment itself is less apparent and comparing the relative magnitudes against the force is difficult. This strategy improves the mechanical efficiency of insertion through experimental validation and provides ergonomic and physiological benefits by reducing the perceived effort when appropriately applied. In practical connector assembly, combining a linear force with a small rotational moment can offer two advantages over vertical pushing alone. First, muscle activation exhibits highly nonlinear behavior; as activation approaches maximum capacity, even minor additional loads significantly increase effort and fatigue^51–53. Therefore, instead of relying solely on small muscle groups for linear force, engaging additional muscles to generate rotational torque can help expand the activated pool and evenly distribute the load. Second, concentrating high activation levels in a single muscle group accelerates fatigue accumulation and reduces muscle endurance. Spreading the workload across multiple muscle groups preserves endurance, sustains performance, and may even improve performance^54,55.

When connector insertion depends solely on axial pushing, the primary pushing muscles often operate near their maximum capacity, making the task feel physically demanding and potentially inefficient. By contrast, a combined force and moment approach can distribute the workload among multiple muscle groups. Although total muscle activation may remain similar, neither the pushing nor the torque-generating muscles necessarily reach their individual fatigue thresholds. This distribution can delay the onset of muscle fatigue, reduce peak strain, and improve comfort and endurance during repeated insertions. For example, the human pinch-grip push capacity with the wrist in a neutral position—common when gripping a small connector and pressing it linearly into a hole—reaches approximately 96.4 and 73.1 N in males and females, respectively, whereas the corresponding maximum torque during wrist flexion-extension averages 2.0–2.5 and 1.5–2.0 Nm. Although the exact forces and torques generated by the human hand and wrist vary based on experimental conditions, measurement setups, and individual differences^56–60, several tight-fitting connectors require insertion forces that approach or surpass the maximum push capability of the human wrist. Relying solely on linear pushing under these circumstances results in significant muscular exertion, fatigue, and discomfort. By contrast, applying a modest rotational moment—just 10–20% of maximal wrist torque—significantly reduces the required force, lessens peak muscle loading, and enhances overall insertion efficiency. We conducted an EMG experiment (Supplementary note 7) to compare four muscle activation patterns between pure pushing and pushing-and-wiggling conditions; the results showed that engaging wrist-rotation muscles can help make the task easier.

Ultimately, our investigation began with the simple observation that many individuals instinctively wiggle connectors during insertion, prompting the question of how a slight rotational moment could aid engagement. Through analytical modeling and experimental validation of the two-leg model, we demonstrated that combining moments with linear force provides clear mechanical and ergonomic advantages. We also acknowledge that real-world connector assemblies involve numerous complexities, including unpredictable contact conditions between non-rigid mating parts, friction effects arising from motion, and bending phenomena. Nevertheless, by focusing on the fundamental mechanics of rotation, our model provides a clear explanation for why integrating moments and linear force reduces insertion effort and enhances performance. Specific cases do exist where the proposed force-moment combined strategy is either less effective or inapplicable. For instance, inserting rigid components such as high-precision steel or aluminum parts makes applying moments virtually impossible due to their inherent lack of compliance and tolerance. Another example involves connectors with exceptionally narrow lateral profiles, where rotational inputs fail to produce significant mechanical advantage because of a limited moment arm. In these scenarios, precise alignment of both position and orientation becomes crucial, and minimal axial oscillations may outperform rotational strategies. Aside from these minor exceptions, the two-leg model remains both valid and robust for most insertion tasks involving tight-fitting components. These findings not only deepen our understanding of human connector manipulation but also provide foundational insights for the design of robotic and automated systems that replicate human-like insertion strategies. As a practical demonstration of its applicability, a series of robotic connector assembly and disassembly tasks with various connector types are provided in the Supplementary Movie, underscoring the model’s potential for enabling safer and more reliable automation.

Methods

Overall system design

This study aims to validate the proposed two-leg model for socket assembly and disassembly by conducting experiments under conditions that eliminate human-induced variability using a custom-designed device. The device was employed to conduct insertion and removal experiments under meticulously controlled force and moment conditions. While the primary experiments utilized the LCB-30 two-pin connector, the device accommodates other connector types by swapping the connector-holding components. Supplemental experimental results for alternative connector types are available in the Supplementary Information. The device consists of two principal sections: an upper mount and a bottom mount. On the upper mount, a vertical force and a wiggling moment are applied to the upper connector part (female connector), which is then assembled with or disassembled from the lower connector part (male connector) secured to the bottom mount. The vertical force is generated either by placing weight disks on the handle’s top seat, which is attached to the upper mount, or by vertically pulling the handle, while the wiggling moment is applied by a DC motor mounted on the upper mount. The bottom mount houses the fixed male connector, two laser displacement sensors positioned to measure the insertion depths of the left and right legs, and an FT sensor that records forces and moments during insertion and removal. A comprehensive description of the device and its setup is provided in the Supplementary note.

Socket displacement measurements

Two upward-facing laser displacement sensors were installed beneath the male connector to measure the insertion depths of the left and right sockets (i.e., the upper connector part in this experimental setup), as shown in Fig. 4. These sensors were positioned symmetrically at lateral offsets from the connector’s centerline, where the lateral offset represents the user-defined distance from the connector’s center to each leg. For a two-pin connector, this corresponds to the lateral distance between the connector center and each pin location. Each sensor records the vertical displacement of the upper socket’s top surface on its respective side (left or right), with displacement calibrated to zero at the initial pin-socket contact. Under purely vertical insertion, perfect alignment between the fixed male pins and female sockets results in identical measurements, denoted as $l_l$ and $l_r$, which directly correspond to the actual vertical displacement of the sockets $L_l$ and $L_r$, respectively. However, when rotational motion is introduced, the sensor outputs no longer accurately reflect the displacements of the left and right sockets. In this scenario, the actual vertical displacement $L_l$ and $L_r$ of each socket relative to the sensor plane is defined by $L_l=X-\mathrm{r}\cdot \sin \theta$, $L_r=X+\mathrm{r}\cdot \sin \theta$, where $X=\left(l_l+l_r\right)/2-h\cdot \sin \theta \cdot \tan \mathrm{\theta }$, where denotes the displacement of the top center of the female part, h denotes the height of the female connector, and r is the half-distance between the two pins.

Static friction force estimation

In the proposed two-leg model, introducing a slight rotational moment to the connector generates linear forces of equal magnitude but opposite direction (moment-induced forces) on both the left and right sides. If the resulting reaction force exceeds the maximum static friction, the leg will move forward or backwards; otherwise, it remains stationary. Thus, comprehending the static friction between male pins and their corresponding female sockets on both sides is crucial. Although manufacturers provide recommended insertion forces, these values serve as conservative minimums and do not consider variables such as pin count, insertion depth, and manufacturing tolerances. Therefore, we experimentally determined the approximate static friction forces on both the left and right sides. To assess static friction as a function of insertion depth, we secured the male and female connector components in our testing apparatus, meticulously aligned them, and initiated initial contact in preparation for insertion. We incrementally added weights to the top seat of the handle, steadily increasing the pressing force applied to the connector. At each specified insertion depth, we recorded the force at which motion first commenced. The collected force–depth data were initially divided by two to estimate the static friction acting on each side. Subsequently, the data were fitted to a polynomial model to characterize the variation of force with insertion depth. For the two-pin connector, the derived values directly correspond to the friction at the left and right pin–socket interfaces. For multipin connectors, the friction required to initiate the movement of each leg corresponds to the combined friction of all pins on either the left or right side of the connector. Because of various uncertainties—including pin contact conditions, housing friction and manufacturing tolerance–induced misalignments generating lateral forces and the velocity dependence of friction—the exact frictional force needed during connector assembly cannot be precisely determined. However, these minor inaccuracies in the friction model are negligible because, under a quasi-static assumption, this study focuses on identifying when a stationary leg begins to move under applied forces and elucidating the relationships and principles connecting those forces to the mating process. Consequently, we use the maximum static friction force of the legs as the boundary thresholds. Details of the static friction modeling are provided in the Supplementary information.

Author contributions

J.C. and S.H.S. conceived the idea for this study. J.P. and J.C. developed the theoretical formulation and performed the analysis. J.P., Y.K., M.K., D.I.P., and J.C. verified the analytical methods. J.P. and Y.K. conceived, planned, and conducted the experiments. J.P. and J.C. wrote the manuscript with support from S.H.S. and D.I.P. J.C. and D.I.P. supervised the study. All authors discussed the results and contributed to the final manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was supported by the Korea Evaluation Institute of Industrial Technology (KEIT) grant funded by the Ministry of Trade, Industry and Energy (MOTIE) (No. 20023257). This work was also partially supported by Korea University.

Data availability

The data generated in this study have been deposited in the Figshare database under accession code 10.6084/m9.figshare.32133736.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74643-3.