Design Methods for Compliant Mechanisms: A Systematic Review Supported by Bibliometric Analysis

Abstract

The design of compliant mechanisms is a multidisciplinary field that integrates structural optimization, kinematics, and materials science to develop systems capable of generating motion through elastic deformation. Over the past six decades, research on compliant mechanisms has grown quickly, encompassing applications ranging from micro- and nano-manipulation to soft robotics and precision engineering. This study presents a comprehensive historical and thematic overview of compliant mechanism design methods through a bibliometric analysis conducted in accordance with the PRISMA methodology. A bias reduction method is proposed for bibliometric analysis, and its limitations are discussed. A total of 10,425 documents published between 1966 and 2025 were retrieved from the Scopus database, revealing an average annual growth rate of 10.35%. The analysis was performed using the Bibliometrix and VOSviewer software packages to conduct performance analysis and science mapping, enabling the identification of influential authors, key publications, and emerging research clusters. The performance analysis results allow, among other things, the identification of the most cited authors and works, which, in turn, facilitates faster identification of the original authors and ideas that gave rise to subsequent thematic branches. Science mapping identified dominant thematic fields underlying past, present, and future trends in compliant mechanisms and their applications. A reader beginning their exploration of the field of compliance mechanisms will find in this work a guide refined by statistical methods, free from the personal bias of its authors. On the other hand, those seeking to understand thematic trends in the field of compliant mechanisms, as well as niche research opportunities, can use the networks and tables generated to explore new possibilities.

1. Introduction

1.1. Overview

When a research field evolves as extensively as compliant mechanisms, tracing the origins of specific subthemes and identifying a coherent critical path for examining their theoretical foundations become increasingly challenging. In such contexts, review articles play a crucial role in organizing accumulated knowledge and mapping the intellectual development of the discipline. Among the defining characteristics of compliant mechanisms, the type of compliance, distributed or concentrated, stands out as a fundamental feature, as it inherently determines the mechanism’s functional capabilities, application potential, and the associated design and analysis methodologies.

It is well established that, although originally developed for structural optimization problems, topology optimization techniques have become the dominant approach for achieving distributed compliance. In contrast, compliant mechanisms based on flexure hinges, often derived from rigid-body kinematic formulations such as the pseudo-rigid-body model (PRBM) [1], are commonly associated with concentrated compliance. The field of compliant mechanisms has now reached a high level of scientific maturity, as reflected in the substantial number of review and state-of-the-art publications. While many of these studies focus on theoretical advances, the exponential growth of literature has also given rise to specialized reviews addressing practical applications across a wide range of domains, including aerospace systems, piezoelectric actuators, metamaterials, precision engineering, and soft robotics.

The present review is therefore particularly timely, as it provides a comprehensive overview of the current state of compliant mechanism design methodologies. Beyond synthesizing existing knowledge, this study establishes a quantitative reference framework that enables the identification of influential authors, seminal contributions, and emerging research trends. The integration of a bibliometric analysis with the PRISMA methodology ensures methodological rigor and systematic transparency, allowing for a detailed mapping of design strategies and application domains that have shaped the evolution of the field. Ultimately, this review captures the dominant trends in compliant mechanism research while highlighting emerging and underexplored areas that offer promising directions for future investigation.

First, a concise historical background is presented to trace the origins of compliant mechanism research and to clarify its principal design concepts, namely, concentrated and distributed compliance. Subsequently, a bibliometric analysis guided by the PRISMA methodology [2] is conducted to ensure transparency and reproducibility. This analysis includes a performance assessment followed by scientific mapping to identify research trends, leading authors, thematic evolution, and other relevant statistical indicators that provide a comprehensive overview of the field and its current challenges. The bibliometric analysis is supported by VOSviewer [3] and the bibliometrix package [4], implemented in the R environment, thereby establishing a robust knowledge structure from which more focused and in-depth reviews can be developed.

1.2. Historical Background

Presenting a historical overview of pioneering work on flexible mechanisms is essential. Despite the lack of metadata in older records, which often results in their exclusion from bibliometric studies, they remain key to understanding the field’s origins. Revisiting these early studies provides the context necessary for evaluating their preliminary hypotheses, limitations, and influence on the field’s evolution.

It is also worth noting that, when new analytical or design methodologies were first introduced, researchers often adopted a cautious approach in defining their theoretical scope and practical applicability. This caution aimed to prevent individual studies from becoming overly ambitious or diffuse. In many cases, unresolved questions arising from these foundational investigations proved sufficiently complex to evolve into distinct research topics or independent lines of inquiry. Consequently, evaluating how the field has progressed in addressing these challenges can be effectively achieved by examining the temporal evolution of the themes identified in this review.

According to Howell [5], one of the most influential and widely cited researchers in the field, a compliant mechanism is a mechanical device that transmits or transforms motion, force, or energy. Unlike rigid-body mechanisms, mobility in compliant mechanisms is achieved wholly or partially through the elastic deflection of flexible members rather than through conventional kinematic joints or pairs. The use of flexible elements in mechanical systems has a long history, with early examples dating back to devices such as bows and catapults. In these systems, potential energy was stored in the elastic deformation of beam-like members, which deflected under applied loads and subsequently released the stored energy to perform a specific task. Although flexible elements have been exploited in mechanical devices for centuries, the development of systematic analysis and design methodologies occurred much later. Historically, such designs relied largely on trial-and-error approaches.

Figure 1 illustrates a conventional four-bar mechanism composed of four single-degree-of-freedom rotational joints (denoted as J₁). In contrast, Figure 2a, presents a compliant equivalent of the same mechanism, in which flexible hinges or flexural joints are employed to replace the conventional rotational joints.

The mechanism shown in Figure 2a deforms primarily in the regions of the notch-type flexure hinges due to their reduced cross-sectional area, which renders these sections more compliant. In contrast, the remaining portions of the mechanism, characterized by larger cross-sectional areas and a higher stiffness, experience negligible deformation and stress. This design approach, in which deformation and stress are intentionally concentrated within localized material regions, is referred to as a compliant mechanism with lumped compliance. By comparison, Figure 2b presents a mechanism that incorporates leaf-type flexural elements, allowing strain to be distributed over a broader material region. Owing to this characteristic, the mechanism is classified as a compliant mechanism with distributed compliance.

In precision engineering, mechanical compliance is mathematically defined as the reciprocal of stiffness, a convention established in the foundational work on flexure hinges by Paros and Weisbord [7] and extensively analyzed in modern literature by Lobontiu [8,9] and Smith [10]. Early research efforts focused almost exclusively on the defining characteristic of compliant mechanisms, namely, the flexible elements themselves. However, this narrow focus soon revealed a conceptual difficulty. In a strict mechanical sense, no component is perfectly rigid, as all bodies experience elastic deformation when subjected to external loads. Consequently, if compliance were interpreted too broadly, a wide range of conventional machine elements (such as gears, cams and followers, springs, belts, and chains) would also need to be classified as compliant elements. This conceptual issue had already been identified by Midha, Her, and Salamon [11,12,13], who proposed a more restrictive definition, characterizing compliant mechanisms as systems composed exclusively of flexible links, thereby excluding conventional machine components.

The study of elastic phenomena and their influence on machine motion and predictability became particularly prominent between the 1950s and 1970s. This interest was largely driven by challenges associated with the dynamic behavior of mechanical components in high-speed machinery and internal combustion engines, where rigid-body assumptions were no longer sufficient for accurate motion prediction. In such systems, elastic deformations significantly affect performance, stability, and reliability.

Within this context, studies by Hain [14] and Livermore [15] investigated the use of springs for load compensation and equilibrium configurations in suspension systems. Additional examples of spring-based mechanisms were documented by Artobolevski [16,17,18], who presented an extensive collection of devices employing elastic elements as functional components. Research by Sieker [19,20] on gearboxes with flexible links, as well as Barkan’s work [21] on high-speed cam-follower valve systems, demonstrated that elastic deformation had to be explicitly considered to accurately predict motion in high-speed applications. Similarly, Meyer zur Capellen [22,23,24,25] analyzed load transmission and vibrational effects in crankshaft mechanisms, incorporating inertial and elastic effects into the analysis.

A significant conceptual shift occurred with the work of Burns [26], which laid the foundations of what is now recognized as compliant mechanism theory. Burns investigated the synthesis and analysis of four-bar mechanisms containing a single flexible link undergoing large elastic deformations. His work is notable for two defining characteristics: the explicit consideration of large deformation behavior and the focus on slow, quasi-static motions (placing the analysis within the framework of kinetostatics). These characteristics closely align with those of earlier studies by Hain [14] and Livermore [15] and represent a departure from purely dynamic elasticity studies.

In parallel, researchers such as Boronkay [27], Davidson [28], Shoup [29], Winfrey [30], and Erdman [31,32] explored the analysis and synthesis of flexible mechanisms from diverse perspectives and for varying objectives. Despite their differences, these studies shared common challenges. As operating speeds increased, the idealization of links as rigid bodies became invalid, and inertial effects coupled with elastic deformations could no longer be neglected. This necessitated the development of mathematical models capable of capturing wave propagation and distributed elasticity effects in flexible links to accurately predict end-effector motion. These early investigations established much of the nomenclature, modeling approaches, and graphical representations that continue to be used in flexible mechanism research. According to Howell, A. Midha should be regarded as the father of compliant mechanisms, as the term “compliant mechanisms” (restricted to mechanical systems composed exclusively of flexible links) was formally introduced by Her, Midha, and Salamon [12,13,33].

Subsequently, Ananthasuresh [34] and Frecker [35] proposed a systematic classification of compliant mechanisms. In this framework, fully compliant mechanisms are defined as systems that do not contain traditional kinematic joints. This category is further subdivided into mechanisms with lumped compliance and those with distributed compliance. In contrast, partially compliant mechanisms retain at least one conventional kinematic joint, with flexibility introduced selectively in specific links. Frecker further distinguished two primary design methodologies: kinematic-based approaches and topology-optimization-based approaches. The former is typically represented by the pseudo-rigid-body model, which consistently produces mechanisms with lumped compliance. The latter encompasses a range of topology optimization techniques that naturally lead to mechanisms with distributed compliance. From this point onward, the evolution of compliant mechanism research followed multiple trajectories, yet most developments can be traced back to these two fundamental design paradigms. The present study does not seek to establish the superiority of one approach over the other but rather to identify the thematic domains that have emerged around them and to illustrate the breadth of their applications.

1.3. Aim of This Work

The present study aims to conduct a comprehensive performance analysis and science mapping of compliant mechanism design methodologies using a bibliometric approach structured in accordance with the PRISMA methodology. The analysis covers an extended time span to capture both the historical evolution and the current research landscape of the field. This work demonstrates the sustained vitality and scientific relevance of compliant mechanism research, reinforcing its position as a dynamic and continuously evolving area of study. In addition, the study identifies a core set of influential publications and leading authors that constitute essential references for understanding past developments, current research trends, and emerging directions. It is noteworthy that, over the last two decades, compliant mechanism research has experienced exponential growth, largely driven by advances in computational design methods, additive manufacturing technologies, and the development of multifunctional and smart materials.

2. Materials and Methods

A traditional systematic review typically includes a collection of works and authors personally selected by the writer. This can introduce bias into the conclusions or their representativeness, as they may be influenced by the co-authors’ backgrounds or personal preferences. A review study, such as the one presented here, supported by bibliometric techniques, acts as a filter to ensure impartiality.

This review was conducted using a bibliometric analysis to synthesize the most relevant and influential research on compliant mechanisms. The term bibliometrics was first introduced by Pritchard [36], who defined it as the application of mathematical and statistical methods to books and other media of communication [37]. To systematically assess research performance and the thematic structure within the field, the present study performs a performance analysis followed by science mapping, in accordance with the methodological recommendations of Donthu [38] and Passas [39]. The bibliometric procedure is guided by the PRISMA methodology [2], thereby ensuring transparency, rigor, and replicability.

Previous studies have demonstrated the effectiveness of bibliometric approaches for analyzing large volumes of scientific information and identifying research trends and thematic structures. In the field of mechanism design, Flores [40,41] and Chen [42] applied bibliometric techniques to examine the evolution of research topics and scientific output. Similarly, García-León [43] and Jaramillo [44] employed a bibliometric analysis to extract statistically supported insights from extensive datasets. Within the specific domain of compliant mechanisms, Huxman [45] and Llanos [46] successfully applied the PRISMA framework to conduct systematic reviews. Moreover, Pham and Lee [47] established a methodological precedent by integrating PRISMA into a bibliometric analysis in the context of expert finding systems, demonstrating the versatility and robustness of this approach. Although their study addressed a different thematic domain, the methodology is directly transferable to engineering research, particularly to the study of compliant mechanisms.

2.1. Bibliometric Analysis Guided by PRISMA

Zupic [48] and Donthu [38] proposed workflow diagrams outlining the general procedure for conducting bibliometric analyses (BAs). While these workflows are effective, they were not explicitly designed to integrate the PRISMA methodology. For this reason, the present study introduces a customized workflow for a bibliometric analysis guided by PRISMA (Figure 3), with specific adaptations to the software tools employed. The workflow begins with the definition of the research topic, which includes establishing the objectives of the bibliometric analysis and formulating research questions. Subsequently, eligibility criteria are defined to specify the characteristics that studies must meet to be included in the analysis. The search strategy then addresses the selection and justification of the bibliometric database, the formulation of the search query in accordance with the review objectives, and the specification of any filters applied. The resulting records are exported in Excel CSV format to enable systematic identification, screening, and study inclusion.

The studies included in the bibliometric analysis are subsequently subjected to a controlled and iterative refinement process designed to minimize potential bias in the review results and conclusions. This process begins by exporting the selected records to bibliometric analysis tools (Bibliometrix (R package version 4.6.1) and VOSviewer (v.1.6.20)) and selecting analytical techniques aligned with the objectives of the review. These tools generate visualizations and quantitative outputs that are carefully examined to detect and correct inconsistencies. Particular attention is given to metadata errors, such as misspelled author names, inconsistent country affiliations, and duplicate records arising from typographical variations. After these inconsistencies are corrected or removed, the refined dataset is reimported into the bibliometric software, and the analysis is repeated until the outputs are free of detectable errors. Only then is the interpretation of results, discussion, and formulation of conclusions undertaken.

Considering the above, this bibliometric analysis seeks to address the following research questions (RQs):

➢

RQ1: How productive and relevant is research on design methods for flexible mechanisms?

➢

RQ2: What are the primary design methodologies for flexible mechanisms?

➢

RQ3: What are the principal applications for flexible mechanisms?

➢

RQ4: Which design techniques, authors, and key works are most recommended when conducting studies in these application areas?

Given the diversity of objectives, no single bibliometric tool is sufficient to address them comprehensively. Therefore, this study combines performance analysis (focused on publication and citation metrics) with science mapping techniques, including citation, co-citation, co-word, and co-authorship analyses.

2.1.1. Eligibility Criteria (BA Data Collection)

The eligibility criteria for including documents in the bibliometric analysis were defined based on several methodological considerations. First, the selected bibliographic database was Scopus. Scopus is one of the largest and most comprehensive databases in the fields of engineering and applied sciences. Comparative studies (such as [49]) have highlighted differences in coverage among Google Scholar, Web of Science, and Scopus. Although Google Scholar generally offers broader coverage, it does not support the structured export of metadata required for post-processing with bibliometric tools such as Bibliometrix [50]. Consequently, Google Scholar is more suitable as an academic search engine rather than as a bibliographic database for a bibliometric analysis [51]. Web of Science remains a highly relevant bibliometric resource; however, in the present case, the search equation designed for this study yielded a substantially higher number of relevant records in Scopus, which constituted the decisive criterion for its selection. Additional comparative analyses of database coverage across different research domains can be found in [52,53,54].

The search strategy was formulated using Boolean operators and consisted of two complementary search equations aligned with the title and objectives of this study, following the search guidelines recommended by Scopus (accessed on 12 May 2025) [55]. The first equation reflects the contemporary terminology associated with compliant mechanisms and was defined as follows: ((compliant W/1 mechanism*) OR (flexible W/1 link*)) AND (design* OR method* OR synthesis OR analysis OR application* OR flexur* OR hinge* OR “topology optimization” OR “topology optimisation” OR distributed OR lumped OR continuum OR compliance). The second equation corresponds to older and more general terminology that historically contributed to the development of compliant mechanisms and related research areas. It was defined as follows: ((mechanism OR (kineto W/1 elastodynamic*)) AND (elastodynamic* OR kineto)) AND (flexible OR synthesis OR linkage*).

Records containing the search terms in the Article Title, Abstract, or Keywords fields were retrieved. Additional filters were applied within the Scopus platform to refine the results, including restrictions to the English language and the limitation of document types to Article, Conference Paper, Book Chapter, Review, Short Survey, and Book. Additionally, a range of years filter from 1966 to 2025 was used.

The search was conducted on 12 May 2026 and yielded 13,168 records in total according to the Scopus platform. These were subsequently exported to Excel CSV format for data preprocessing, where we could immediately see a discrepancy between the number of results reported by the Scopus platform and the number of rows in the downloaded CSV file, which was 13,213; this was considered the initial number of records.

Following the PRISMA methodology and its official guidelines [56], a flow diagram was generated to document the study identification and selection process through databases and registers, as illustrated in Figure 4. The diagram was generated using dedicated PRISMA tools [57,58]. At this stage, 13,213 records were initially identified from the Scopus database. Despite restricting the Scopus results to English, when reviewing the downloaded Excel document and filtering the “Language of Original Document” column, 2 non-English documents were removed (1 in Chinese and 1 in Slovenian) and 90 documents without a language declared. Another 288 records were removed due to duplication when using conditional formatting on the “Title” column, and the dataset comprised 12,833 records eligible for screening.

The resulting CSV file was imported into Biblioshiny to identify the most globally cited documents (those with the highest citation counts across databases) and most locally cited documents (those most frequently cited within the retrieved dataset). The top 250 records from each category were examined. For documents whose relevance to compliant mechanisms could not be readily inferred from authorship or publication venue, abstracts were reviewed to confirm thematic alignment. This screening process led to the exclusion of 114 additional records deemed unrelated to the field after the first filter was applied and another 10 after the second filter was applied, leaving 12,709 records for retrieval.

The selection of a 250-record threshold follows the default setting in Biblioshiny and is generally sufficient to ensure robust graphical visualizations and network analyses.

In PRISMA-guided systematic mappings, the retrieval substage typically includes verification of record accessibility and suitability for inclusion. In the context of this bibliometric study, records lacking essential metadata such as author full name, author keywords, or indexed keywords are not useful, resulting in the exclusion of 9, 0, and 0 records, respectively, thereby leaving 12,700 documents. Finally, while the screening phase in traditional PRISMA applications often focuses on excluding clinical or experimental studies, its adaptation to an engineering-oriented bibliometric review required a modified procedure. As summarized in Figure 4, this customized workflow refined the dataset to include only records that make meaningful contributions to the state of the art in compliant mechanism design.

The final eligibility assessment was conducted as summarized in

Table 1. Results obtained after cleaning the data.

Keyword Used	Records Filtered by “Author Keywords”	Records Related to Compliant Mechanisms Based on “Author Keywords”	New Records Filtered by “Index Keywords”	New Records Related to Compliant Mechanisms Based on “Index Keywords”	Total Records Filtered by Keywords	Records Removed After Review	Reference
Animal Cell	1	0	118	0	119	119	---
Nuclear	25	3	285	4	310	303	[59,60,61,62,63,64,65]
Protein	222	0	618	1	840	839	[66]
Recombination	0	0	1	0	1	1	---
Peptide	26	0	26	0	52	52	---
Mice	2	0	24	5	26	21	[67,68,69,70,71]
Mouse	1	1	3	1	4	2	[72,73]
Dna	33	4	46	3	79	72	[74,75,76,77,78,79]
Binding	6	0	31	0	37	37	---
Chemical	24	0	274	35	298	263	[80,81,82,83,84,85]
Cell	81	33	156	88	236	116	[86,87,88,89,90]
Affinity	0	0	3	0	3	3	---
Molecular	43	2	107	18	150	130	[91,92,93,94,95]
Carboxy	5	0	19	0	24	24	---
Amino	2	0	10	0	12	12	---
Escherichia	2	0	4	2	6	4	[96,97]
Cancer	2	1	6	3	8	4	[98,99,100,101]
Metabolism	1	0	8	1	9	8	[102]
Drug	10	0	29	5	39	34	[103,104,105]
Blockchain	51	0	7	0	58	58	---
Security	64	0	69	1	133	132	[106]
Privacy	30	0	11	0	41	41	---

. This procedure was independently performed by three researchers involved in the study, and the results obtained by each evaluator were subsequently cross-verified to ensure consistency and accuracy. Following this validation process, 2275 records were excluded, resulting in a final dataset of 10,425 entries for subsequent analysis.

• A list of keywords was compiled from the Scopus platform that, in the authors’ judgment, had a low probability of being genuinely related to compliant mechanisms.
• These keywords were filtered in the “Author Keyword” column.
• Subsequently, the results filtered by author keyword were further screened in the “Title” column for terms such as “compliant” or “mechanism”.
• Results from the previous step that did not address compliant mechanisms or their applications were then checked manually by reviewing their abstracts to decide whether they should be excluded from the selection.
• Steps 1 to 3 were repeated, beginning with the “Index Keywords” column.

It is important to note that the PRISMA methodology was originally developed as a framework for conducting systematic reviews in the health sciences, particularly for the evaluation of health interventions, and it was later extended to social and educational research contexts [107]. A health intervention is generally defined as any activity intended to improve human health by preventing, treating, or reducing the severity of disease [108]. Studies in this domain typically report outcomes derived from questionnaires, clinical observations, or randomized experimental measurements applied to individuals. Depending on the study design and implementation, such procedures may introduce various sources of bias that affect the interpretation and predictability of results.

These characteristics imply that several items in the PRISMA checklist are specifically tailored to intervention-based studies and may not be directly applicable to bibliometric reviews. In bibliometric research, the primary data consist of large volumes of metadata, which are processed to generate tables and network visualizations that support the interpretation of scientific trends and relationships. Although the bibliometric workflow generally includes metadata preprocessing steps (such as standardizing author names and removing duplicate records), a residual risk remains that irrelevant documents, duplicated entries, or references containing typographical variations may persist due to author input errors or database inconsistencies.

The likelihood of such inconsistencies increases substantially when analyzing datasets spanning long time periods, as is the case in the present study. Nevertheless, focusing on the most informative and representative elements extracted from bibliometric tables and visualizations is generally sufficient, since network maps and collaboration graphs tend to lose interpretative clarity beyond approximately 100 to 200 nodes. To address these limitations, Figure 3 introduces a conditional feedback loop designed to reduce the probability of erroneous outcomes and misinterpretations in bibliometric analyses while maintaining conceptual alignment with the PRISMA framework and mitigating the risk of residual data errors. This adaptation broadly corresponds to items 11, 18, and 21 of the PRISMA checklist. Strictly speaking, the bias reduction strategy illustrated in Figure 3 does not eliminate bias entirely; however, it substantially minimizes its impact by emphasizing the most representative and influential elements of the dataset. Given that bibliometric analyses provide a macroscopic perspective of a research field based on its most significant features, it is reasonable to assume that the overall conclusions are unlikely to be affected by a limited number of minor uncorrected cases, which rarely attain sufficient prominence to influence the top-ranked results presented in the tables and figures.

2.1.2. Bibliometric Techniques and Software Tools

In accordance with the research questions formulated to support the objectives and scope of this study (namely, performance analysis and science mapping), a range of bibliometric techniques is employed. These include productivity indicators, total citation counts, and average citations per document. Based on these metrics, several bibliometric indices are calculated, including the collaboration index, h-index, g-index, and i-index. Subsequently, science mapping is conducted through co-citation analysis, bibliographic coupling, co-word analysis, and co-authorship analysis, supported by visual representations such as network and clustering maps.

The bibliometric analysis is carried out using the Bibliometrix package [4] and VOSviewer [3]. Comparative studies [109,110] have evaluated the most commonly used tools for bibliometric analysis, examining their features, as well as the bibliographic databases most frequently employed. Based on these comparisons, the selection of Bibliometrix and VOSviewer is well justified, as both tools provide the full range of functionalities required to conduct the comprehensive bibliometric analysis presented in this work. In addition, both tools are fully compatible with data retrieved from Scopus, are freely available, and generate diverse and informative visual outputs that clearly reveal relevant relationships and structural patterns within the data.

Several studies have contributed to establishing Bibliometrix as a robust and accessible bibliometric tool. For example, the works of Büyükkidik [111] and Barros et al. [112] can be regarded as introductory tutorials on its use. Furthermore, Bibliometrix has been applied across a wide range of research fields in studies such as [4,113,114,115,116,117], often with the direct involvement of its developers. Collectively, these contributions form a valuable methodological foundation for generating and interpreting bibliometric indicators and visualizations. In addition, studies by Sahoo [118] and Kumar [119] demonstrate that the combined application of Bibliometrix and VOSviewer can effectively support comprehensive bibliometric analyses.

3. Results and Discussion

3.1. Performance Analysis

Performance analysis evaluates the scientific output of institutions, researchers, countries, and publication sources in order to assess their influence within a specific research domain. This evaluation is based on a range of bibliometric indicators (as discussed in [38]) and is implemented in the present study following the workflow illustrated in Figure 3. The results obtained here directly address RQ1, clearly identifying the level of productivity by author, country, and source. The level of collaboration between countries is analyzed, revealing the relevance of the research topic.

3.1.1. Publication-Related Metrics

A summary of the bibliometric indicators generated using Bibliometrix is presented in

Table 2. BA main results.

Description	Results
Timespan	1966:2025
Sources (Journals, Books, etc.)	2665
Documents	10,425
Annual Growth Rate %	12.02
Document Average Age	12.8
Average Citations per Doc	18.29
References	358,843
Document Contents
Keywords Plus (ID)	37,309
Author’s Keywords (DE)	16,200
Authors
Authors	19,139
Authors of Single-Authored Docs	515
Author Collaboration
Single-Authored Docs	686
Co-Authors per Doc	3.4
International Co-Authorships %	15.12
Document Types
Article	5849
Book	12
Book Chapter	196
Conference Paper	4298
Review	69
Short Survey	1

. The results indicate an average annual growth rate of 12.02% in scientific publications over the period of 1966–2025. As illustrated in Figure 5, both the annual and cumulative publication trends indicate that the field has not yet reached a saturation stage, suggesting sustained and increasing research interest in compliant mechanism design.

According to Table 2, the final bibliometric dataset comprises a total of 10,425 publications (TP) authored by 19,139 contributors (NCA). Of these, 515 researchers produced 686 single-authored documents (SA). Over the full 59-year observation period (NAY), the productivity per active year (PAY) is calculated as 176.69 publications per year. When the analysis is limited to the continuous period of scientific production beginning in 1980, the number of active years decreases to 45, while the PAY increases to 231.15 publications per year, reflecting a sustained and accelerating growth trend.

Figure 6 presents the 20 most productive publication sources in which the retrieved works appeared. Notably, the first six sources are of a more general scope, primarily covering mechanism theory, mechanical design, and robotics. While these journals and conference proceedings include a significant number of applied and high-impact contributions related to compliant mechanisms, they are not specifically dedicated to niche sectors or application-oriented subfields. In contrast, the subsequent sources in the ranking (such as Precision Engineering, Structural and Multidisciplinary Optimization, Robotics and Automation Letters, Smart Materials and Structures, Proceedings of the American Control Conference, and Lecture Notes in Computer Science) exhibit a clearer specialization, addressing targeted themes and advanced applications within compliant mechanism research.

Based on the source titles and their corresponding publication counts, this analysis enables the identification of the most reputable journals and conferences in the field while simultaneously providing an initial overview of the dominant specialized research topics. Whereas the increasing productivity trend illustrated in Figure 5 reflects the overall vitality and sustained growth of compliant mechanism research, Figure 6 specifically highlights the most productive publication venues. This information is particularly valuable for guiding future researchers toward the journals and conferences that are most relevant for disseminating new findings in this domain.

Figure 7 presents the 25 most productive authors in the field of compliant mechanisms. Larry Howell leads the ranking with 173 publications, followed by Zhang Xianmin with 164. Authors occupying the highest positions are typically senior researchers and leaders of well-established research groups, whose academic trajectories warrant closer examination. These contributors have played a central role in shaping and consolidating the theoretical and methodological foundations of the field, often producing a substantial body of work focused on closely related research themes.

For instance, Larry Howell, together with Ashok Midha, is widely recognized for the development and dissemination of the pseudo-rigid-body model (PRBM), which has become a cornerstone in the analysis and synthesis of compliant mechanisms. In parallel, researchers such as Zhang Xianmin, Mary Frecker, and G. K. Ananthasuresh have made seminal contributions to the advancement of continuum topology optimization approaches, significantly expanding their application in compliant mechanism design.

Figure 8 presents the countries with the highest levels of scientific productivity in compliant mechanism research. China and the United States emerge as the clear leaders, followed by India, Germany, Japan, Italy, Canada, and the UK. While China currently occupies the top position in terms of total publication output, Figure 9 illustrates the temporal evolution of national productivity and reveals a dynamic shift in global leadership over time.

The United States dominated publication output from the beginning of the study period until 2020 when China overtook it. The pronounced upward trajectory in Chinese publication output reflects a rapid expansion and sustained investment in research on compliant mechanisms. This trend highlights not only the increasing research capacity of Chinese institutions but also the country’s growing scientific influence and long-term commitment to advancing this field.

3.1.2. Citation-Related Metrics

Figure 10 presents both the average number of citations per year and the cumulative citations received over time. This indicator is useful for identifying periods in which highly influential publications emerged, particularly during early stages of the field when overall scientific output was relatively low, such as the years prior to the 1980s. However, this metric must be interpreted with caution as publication volume increases, since the average number of citations may be disproportionately influenced by a small number of exceptionally highly cited works.

Special attention is typically given to years in which the average number of citations exceeds the total number of publications, as this condition often indicates the presence of seminal contributions. Nevertheless, relying solely on this criterion may introduce bias in later periods that do not satisfy this condition, due to the skewing effect of citation outliers, as previously reported in the literature [120].

By jointly examining citation and publication metrics, it is possible to identify specific time intervals characterized by an unusually high citation impact relative to the number of published works. As illustrated in Figure 10, periods such as 1970–1975 and 1978–1988 stand out and merit closer examination of the documents published during these years, as they are likely to contain foundational and highly influential contributions to the development of compliant mechanism research.

3.1.3. Citation and Publication-Related Metrics

Figure 7 presents the most prolific authors in terms of publication output; however, productivity alone does not adequately reflect the scientific impact or influence of individual contributions. A more informative assessment is provided in

Table 3. Top relevant authors by h-index.

Author	h_Index	g_Index	m_Index	Total Citations	Number of Publications	PY_Start
Howell Larry L.	44	81	1.257	7385	173	1992
Xu Qingsong	32	53	1.455	2984	95	2005
Zhang Xianmin	31	44	1.000	2828	164	1996
Kota Sridhar	27	61	0.818	3729	67	1994
Ananthasuresh G.K.	25	56	0.758	3246	85	1994
Hao Guangbo	25	37	1.471	1678	89	2010
Herder Just L.	25	37	0.926	1601	83	2000
Magleby Spencer P.	25	49	0.926	2543	81	2000
Ling Mingxiang	24	37	2.182	1441	55	2016
Shirinzadeh Bijan	24	42	1.091	2255	42	2005
Sigmund Ole	24	26	0.800	7324	26	1997
Chen Guimin	23	47	1.045	2986	68	2005
Su Hai-Jun	23	38	1.000	1531	58	2004
Tian Yanling	23	46	1.211	2164	46	2008
Feliu Vicente	22	37	0.564	1543	63	1988
Frecker Mary	21	47	0.677	2362	91	1996
Zhu Benliang	20	30	1.333	991	58	2012
Jensen Brian D.	19	37	0.633	1404	39	1997
Korayem M.H.	19	29	0.543	890	34	1992
Midha Ashok	19	50	0.413	2505	62	1981
Zhang Dawei	18	29	1.000	1817	29	2009
Li Yangmin	17	39	0.773	1597	44	2005
Liu Jinkun	17	42	1.133	1101	32	2012
Saxena Anupam	17	34	0.586	1208	48	1998
Wang Michael Yu	17	27	0.739	1624	27	2004

, which relates publication volume to citation performance through established bibliometric indicators, including the h-index, g-index, and m-index. These metrics offer a more nuanced perspective than simple productivity rankings, as they highlight authors whose work, although less extensive in quantity, has exerted substantial influence within the field. For example, Ole Sigmund ranks among the most highly cited researchers despite having only 26 publications, underscoring the exceptional impact of his contributions to topology optimization and its application to compliant mechanism design.

Additionally, the data presented in Table 3 show that Larry Howell achieves both the highest productivity (173 publications) and the largest citation count (7385), reflecting his central role in the development of the pseudo-rigid-body model (PRBM) and in advancing the theoretical foundations of compliant mechanism research. Xu Qingsong and Zhang Xianmin also demonstrate strong citation performance (h = 32 and h = 31, respectively), indicating sustained and influential contributions, particularly in the areas of micromanipulation systems and precision mechanisms. In contrast, authors such as Hao Guangbo (m = 1.471) and Ling Mingxiang (m = 2.182) exhibit comparatively high m-index values, suggesting rapidly expanding research activity and the emergence of new leadership within the field.

Although Figure 11 presents a hierarchy based on publication productivity, it also identifies the most productive years (indicated by point size) and the periods of highest impact (indicated by color intensity) for most of the authors listed in Table 3. This visualization therefore provides a temporal perspective on both research activity and scientific influence.

In addition, Figure 11 highlights emerging influential authors, such as Ling Mingxiang, who began publishing in 2016 and has since accumulated 55 publications. Beyond productivity, his work demonstrates a strong impact, as reflected by a high h-index and the highest m-index among the authors in this list, indicating a rapid and high-quality research trajectory when compared with more established, long-standing researchers. The figure also enables the identification of well-established authors with sustained research activity, including Howell, Frecker, and Saxena. Conversely, it reveals researchers whose publication activity appears to have declined or ceased in recent years, such as Kota, Ashok Midha, and Ananthasuresh, suggesting retirement or a temporary shift away from active research in this domain.

According to Table 2, the dataset exhibits an average of 3.4 co-authors per document, reflecting a mature and well-established collaboration network within the field.

Table 4. Most relevant countries according to number of publications, as well as single and multiple collaboration indicators.

Country	Articles	SCP	MCP	MCP %
China	2145	1835	310	14.5
USA	903	784	119	13.2
India	380	348	32	8.4
Italy	267	223	44	16.5
Canada	223	193	30	13.5
Germany	209	174	35	16.7
Japan	208	182	26	12.5
Iran	185	161	24	13.0
Korea	169	147	22	13.0
United Kingdom	143	99	44	30.8
Spain	127	95	32	25.2
Turkey	112	96	16	14.3
Netherlands	111	92	19	17.1
Australia	107	68	39	36.4
France	100	74	26	26.0
Malaysia	72	48	24	33.3
Singapore	72	45	27	37.5
Switzerland	65	46	19	29.2
Brazil	64	45	19	29.7
Ireland	63	36	27	42.9

reports the 20 countries associated with the authors who have produced the highest number of publications, together with their Single-Country Publications (SCPs) and Multiple-Country Publications (MCPs). These results are also visually summarized in Figure 12.

Countries with high MCP values but relatively low MCP percentages, such as China and the United States, can be regarded as scientific powerhouses in compliant mechanism research, as their large research output is predominantly driven by strong domestic research networks. In contrast, countries with fewer total publications but high MCP percentages reflect pioneering research efforts supported by intensive international collaboration, highlighting their integration into global research networks despite more limited national production. These collaboration patterns provide additional insight into the structural maturity and internationalization of compliant mechanism research. Countries characterized by high domestic productivity and lower relative MCP percentages tend to host consolidated research groups with long-term funding, stable institutional support, and established academic traditions in mechanism theory and mechanical design. Conversely, nations with higher MCP ratios often rely on international partnerships to access specialized expertise, advanced infrastructure, or complementary methodologies, which can accelerate knowledge transfer and innovation. This collaborative behavior is particularly relevant in emerging research regions, where international cooperation serves as a catalyst for scientific visibility and integration into the global research landscape. Overall, the observed balance between national consolidation and international collaboration underscores the globalized nature of compliant mechanism research and reflects its evolution toward a highly interconnected and multidisciplinary field.

3.2. Science Mapping

3.2.1. Citation Analysis

The citation analysis addresses RQ2 and aims to identify the most influential publications in the field by examining the number of citations accumulated over time. The underlying premise is that frequently cited works introduced seminal or transformative contributions at the time of publication, subsequently becoming foundational references across multiple thematic areas. Table 5 summarizes the 20 most cited documents among the 10,425 records analyzed in this study, distinguishing between local citations (within the present dataset) and global citations (across the entire Scopus database).

The results show that Sigmund [121] occupies a central position in the citation landscape, with 404 local citations and 1438 global citations. This prominence consolidates his contribution to topology optimization for compliant mechanisms as one of the most influential theoretical pillars of the field. Other highly cited works include Howell [1] and Frecker [122], which established key methodological frameworks for compliant mechanism design based on flexural pivots and multi-objective optimization, respectively. Works such as those by Cannon [123] and Siciliano [124] reveal the interest in the control of flexible manipulators.

Studies on topological optimization and its applications to the synthesis of flexible mechanisms, such as those carried out by Nishiwaki [125], Pedersen [126], Bruns [127], Xu [128], Kota [129], and Wang [130], are evidence of the strong interest within the scientific community in studying and improving this technique in the design of flexible mechanisms.

Selecting flexible mechanisms for MEMS design applications has been a frequent choice of the scientific community, thanks to the ease of manufacturing them monolithically; the works of Kota [131] and Ananthasuresh [132] in this table are proof of this.

Table 5. Most cited publications.

Author/Year	Title	Local Citations	Global Citations	Ref.
Sigmund O, 1997	On the Design of Compliant Mechanisms Using Topology Optimization	404	1438	[121]
Howell L.L., 1994	A Method for the Design of Compliant Mechanisms with Small-Length Flexural Pivots	286	552	[1]
Frecker M.I., 1997	Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization	232	546	[122]
Howell L.L., 1995	Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms	228	431	[133]
Howell L.L., 1996	Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms	165	307	[134]
Cannon R.H., 1984	Initial Experiments on the End-Point Control of a Flexible One-Link Robot	157	876	[123]
Nishiwaki S., 1998	Topology optimization of compliant mechanisms using the homogenization method	150	396	[125]
Siciliano B, 1988	A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators	147	443	[124]
Howell L.L., 1996	A Loop-Closure Theory for the Analysis and Synthesis of Compliant Mechanisms	147	231	[135]
Pedersen C.B.W., 2001	Topology synthesis of large-displacement compliant mechanisms	146	492	[126]
Bruns T.E., 2001	Topology optimization of non-linear elastic structures and compliant mechanisms	143	1331	[127]
Hopkins J.B., 2010	Synthesis of multi-degree of freedom, parallel flexure system concepts via Freedom and Constraint Topology (FACT)—Part I: Principles	133	321	[136]
Lobontiu N., 2003	Analytical model of displacement amplification and stiffness optimization for a class of flexure-based compliant mechanisms	128	344	[137]
Xu Q., 2011	Analytical modeling, optimization and testing of a compound bridge-type compliant displacement amplifier	100	278	[128]
Kota S., 1995	Designing compliant mechanisms	100	247	[129]
Kota S., 2001	Design of Compliant Mechanisms: Applications to MEMS	98	209	[131]
Ananthasuresh G.K., 1994	Strategies for systematic synthesis of compliant mems	97	193	[132]
Midha A., 1994	On the Nomenclature, Classification, and Abstractions of Compliant Mechanisms	96	166	[138]
Benosman M., 2004	Control of flexible manipulators: A survey	95	353	[139]
Wang F., 2011	On projection methods, convergence and robust formulations in topology optimization	94	1715	[130]

Table 5. Most cited publications.

Author/Year	Title	Local Citations	Global Citations	Ref.
Sigmund O, 1997	On the Design of Compliant Mechanisms Using Topology Optimization	404	1438	[121]
Howell L.L., 1994	A Method for the Design of Compliant Mechanisms with Small-Length Flexural Pivots	286	552	[1]
Frecker M.I., 1997	Topological Synthesis of Compliant Mechanisms Using Multi-Criteria Optimization	232	546	[122]
Howell L.L., 1995	Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms	228	431	[133]
Howell L.L., 1996	Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms	165	307	[134]
Cannon R.H., 1984	Initial Experiments on the End-Point Control of a Flexible One-Link Robot	157	876	[123]
Nishiwaki S., 1998	Topology optimization of compliant mechanisms using the homogenization method	150	396	[125]
Siciliano B, 1988	A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators	147	443	[124]
Howell L.L., 1996	A Loop-Closure Theory for the Analysis and Synthesis of Compliant Mechanisms	147	231	[135]
Pedersen C.B.W., 2001	Topology synthesis of large-displacement compliant mechanisms	146	492	[126]
Bruns T.E., 2001	Topology optimization of non-linear elastic structures and compliant mechanisms	143	1331	[127]
Hopkins J.B., 2010	Synthesis of multi-degree of freedom, parallel flexure system concepts via Freedom and Constraint Topology (FACT)—Part I: Principles	133	321	[136]
Lobontiu N., 2003	Analytical model of displacement amplification and stiffness optimization for a class of flexure-based compliant mechanisms	128	344	[137]
Xu Q., 2011	Analytical modeling, optimization and testing of a compound bridge-type compliant displacement amplifier	100	278	[128]
Kota S., 1995	Designing compliant mechanisms	100	247	[129]
Kota S., 2001	Design of Compliant Mechanisms: Applications to MEMS	98	209	[131]
Ananthasuresh G.K., 1994	Strategies for systematic synthesis of compliant mems	97	193	[132]
Midha A., 1994	On the Nomenclature, Classification, and Abstractions of Compliant Mechanisms	96	166	[138]
Benosman M., 2004	Control of flexible manipulators: A survey	95	353	[139]
Wang F., 2011	On projection methods, convergence and robust formulations in topology optimization	94	1715	[130]

The sustained citation activity of these works confirms their long-term relevance and cross-disciplinary impact, bridging mechanical design, robotics, materials science, and computational optimization within a unified research framework. In addition, the citation analysis performed using Bibliometrix includes the evaluation of locally cited references, a feature that is particularly valuable for identifying seminal works that may not be directly associated with the search keywords or fully indexed within the selected database. Despite this, such studies are repeatedly cited by the research community, underscoring their status as fundamental references and, in some cases, as canonical or introductory sources for the study of compliant mechanisms and their associated design and analysis methodologies.

Table 6. References with most local citations.

Author/Year	Title	Local Citations	Ref.
Howell L.L., 2001	Compliant Mechanisms	1270	[5]
Howell L.L., 2013	Handbook of Compliant Mechanisms	250	[142]
Lobontiu N. 2002	Compliant Mechanisms: Design of Flexure Hinges	179	[8]
Dwivedy S.K., 2006	Dynamic analysis of flexible manipulators, a literature review	138	[143]
Bendsoe M., 2003	Topology Optimization: Theory, Methods, and Applications	118	[144]
Sigmund O., 1997	On the design of compliant mechanisms using topology optimization mechanics of structures and machines	112	[121]
Smith S.T., 2000	Flexures: Elements of Elastic Mechanisms	111	[10]
Lobontiu N. 2003	Compliant Mechanisms: Design of Flexure Hinges	99	[8]
Howell L., 2013	Compliant Mechanisms: 21st Century Kinematics	73	[145]
Meirovitch L.	Analytical Methods in Vibrations	64	[146]

lists the ten most locally cited references, representing the core literature that has exerted the strongest intellectual influence within the retrieved dataset. In contrast, Table 7 presents the ten most influential “hot papers”, which correspond to recent publications that have rapidly accumulated citations compared to others in the same research field. This classification reflects works whose scientific quality, novelty, and relevance were promptly recognized by the academic community, resulting in accelerated citation growth shortly after publication [140,141].

The data in Table 6 confirm the enduring relevance of several foundational texts that continue to shape the theoretical and methodological landscape of compliant mechanisms. Notably, Howell’s 2001 monograph Compliant Mechanisms emerges as the most locally cited reference, with 1270 citations, firmly establishing its status as the seminal and definitive work in the field [5]. Likewise, Lobontiu’s 2002 book Design of Flexure Hinges [8] (179 citations) and the Handbook of Compliant Mechanisms by Howell, Magleby, and Olsen [142] (250 citations) provide comprehensive and systematic frameworks that have become standard references in both academic research and engineering practice.

Following Howell’s seminal work, the two books by Lobontiu and the one by Smith reveal a wealth of research on problems related to the design of flexible hinge-based mechanisms. Sigmund’s and Bendsoe’s works, for their part, reflect the growing body of work on topological optimization applied to the design of flexible mechanisms, while Dwivedy’s work supports the increasing development surrounding flexible manipulators.

Collectively, the most locally and “hot” cited publications delineate the evolutionary trajectory of compliant mechanism research, illustrating how classical theoretical foundations have been progressively integrated with advanced optimization techniques, flexible manipulators, additive manufacturing processes, and the emergence of intelligent and multifunctional materials.

Table 7 presents the ten most influential “hot papers,” defined as publications that have accumulated citations rapidly within a relatively short period after publication, indicating early recognition and high scientific relevance. Pedersen, Buhl, and Sigmund [126] stand out for their pioneering work on large-displacement topology synthesis, which remains a cornerstone of computational design optimization in compliant mechanisms. Similarly, Howell and Midha [133] introduced parametric deflection models that underpin the pseudo-rigid-body method, while Bourdin [147] proposed filtering techniques that significantly improved numerical stability and robustness in topology optimization algorithms.

More recent contributions, such as Wang and Xu [148] and Chen et al. [149], reflect the contemporary evolution of the field by addressing bistable mechanisms and constant-force systems, respectively—applications that are increasingly relevant to soft robotics, precision engineering, and adaptive devices. The inclusion of Burns [26] among the most influential works is particularly noteworthy, as it highlights one of the earliest theoretical explorations of flexible link synthesis. This result underscores the enduring impact of classical mechanical principles on the development of modern compliant mechanism design methodologies.

Table 7. Top 10 hot papers.

Author/Year	Reference	Local Citations	Ref.
Howell L 1995	Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanisms	20	[133]
Bourdin B., 2001	Filters in Topology Optimization	17	[147]
Pedersen C.B.W., 2001	Topology Synthesis of Large-Displacement Compliant Mechanisms	17	[126]
Burns R., 1965	The Kinetostatic Synthesis of Flexible Link Mechanisms	13	[26]
Howell L 1996	Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms	12	[134]
Wang P., 2018	Design and Modeling of Constant-Force Mechanisms: A Survey	12	[148]
Chen Q., 2019	Topology Optimization of Bistable Mechanisms with Maximized Differences Between Switching Forces in Forward and Backward Direction	12	[149]
Howell L., 1995	Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanism	11	[133]
Pucheta M.A., 2010	Design of Bistable Compliant Mechanisms Using Precision-Position and Rigid-Body Replacement Methods	11	[150]
Rahimi H.N 2014	Dynamic Analysis and Intelligent Control Techniques for Flexible Manipulators: A Review	10	[151]

Table 7. Top 10 hot papers.

Author/Year	Reference	Local Citations	Ref.
Howell L 1995	Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanisms	20	[133]
Bourdin B., 2001	Filters in Topology Optimization	17	[147]
Pedersen C.B.W., 2001	Topology Synthesis of Large-Displacement Compliant Mechanisms	17	[126]
Burns R., 1965	The Kinetostatic Synthesis of Flexible Link Mechanisms	13	[26]
Howell L 1996	Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms	12	[134]
Wang P., 2018	Design and Modeling of Constant-Force Mechanisms: A Survey	12	[148]
Chen Q., 2019	Topology Optimization of Bistable Mechanisms with Maximized Differences Between Switching Forces in Forward and Backward Direction	12	[149]
Howell L., 1995	Parametric Deflection Approximations for End-Loaded Large-Deflection Beams in Compliant Mechanism	11	[133]
Pucheta M.A., 2010	Design of Bistable Compliant Mechanisms Using Precision-Position and Rigid-Body Replacement Methods	11	[150]
Rahimi H.N 2014	Dynamic Analysis and Intelligent Control Techniques for Flexible Manipulators: A Review	10	[151]

The citation analyses presented in Table 5, Table 6 and Table 7 provide a comprehensive intellectual roadmap of the evolution of compliant mechanism research. They clearly delineate the transition from early foundational theoretical models to contemporary computational strategies and application-oriented developments, highlighting how seminal conceptual contributions continue to shape and inspire current innovations in the field. These results confirm the cumulative and integrative nature of compliant mechanism research, in which classical analytical frameworks coexist with advanced numerical and design methodologies.

In this context, Bibliometrix proved to be a particularly valuable tool, as its reference analysis module enabled the systematic inclusion of gray literature, such as theses and scholarly books. These sources often encapsulate critical theoretical insights, design methodologies, and historical perspectives that are not always fully represented in peer-reviewed journal articles yet play a fundamental role in the consolidation and transmission of knowledge within the discipline.

3.2.2. Co-Citation Analysis

By assuming that two or more publications cited together in a given work share a common thematic orientation, co-citation analysis aims to reveal the intellectual structure of the field under investigation and to identify its seminal contributions. The frequency with which specific references are cited, combined with their co-citation patterns, enables the identification of highly influential works within distinct subthemes, positioning them as central nodes within clusters in network visualizations. Gmür [152] defines a “cluster” as a group of references that exhibit multiple interconnections, as determined by the linkage rules of the co-citation network.

The paper-based co-citation network is presented in Figure 13, which helped identify the foundational paradigms of compliant mechanisms according to RQ2, while Table 8 lists the three most representative documents associated with each identified cluster. The unit of analysis was based on the cited references, with a minimum of 20 citations per cited reference and a total of 50 nodes.

The resulting co-citation map reveals several well-defined thematic clusters that reflect the core research directions in compliant mechanisms. These include foundational studies on topology optimization and its application to continuum compliant mechanism synthesis (red cluster); mechanisms based on flexible joints and their modeling, design, and applications (blue cluster); nonlinear modeling of large deflections in flexible links (green cluster); and design methodologies grounded in the pseudo-rigid-body model (purple cluster). In addition, the network highlights researchers who have played a pivotal role in bridging theoretical domains and integrating distinct research streams. As illustrated in Figure 13, key contributors in this regard include Smith, Hopkins, Bendsøe, Lobontiu, and Howell.

A closer examination of the clusters identified in the co-citation network provides deeper insight into the intellectual foundations and internal structure of compliant mechanism research. Each cluster represents a coherent body of knowledge shaped by shared theoretical principles, modeling strategies, and application domains.

The resulting co-citation map reveals several well-defined thematic groups that reflect the main research directions in compliant mechanisms. The red group focuses on applying topological optimization techniques to the design of flexible distributed compliance mechanisms. This cluster includes the mathematical origins of structural optimization and the homogenization method proposed by Bendsoe and Kikuchi [153]. Structural optimization techniques represented a disruptive advance for structural design, but they implied incompatibilities with the principle of a flexible mechanism that, paradoxically, should be as flexible as possible when deforming but extremely rigid when interacting with or manipulating other objects. This dilemma was resolved with the multi-objective formulation of Frecker et al. [122,125]. Sigmund and Bendsoe [154] formulated the SIMP (Solid Isotropic Material with Penalization) method. Once the potential of optimization techniques in the design of flexible mechanisms was demonstrated, research such as that of Bruns [127] and Pedersen [126] suggested the next objectives to pursue: the integration of nonlinear FEA for large deformations.

The blue cluster comprises works that aim to design lumped compliance mechanisms primarily based on flexible hinges. Two theoretical approaches stand out here: the formulation based on Howell’s pseudo-rigid-body method [1], which replaces traditional rotational joints with torsion springs whose spring constant and deflection are associated with short-length flexible hinges, and the approach led by Lobontiu [8], which, instead of drawing an analogy between flexible hinges and torsion springs, focuses on the precise formulation of beam-like flexible elements using Euler–Bernoulli and Timoshenko theory. This allows him to formulate the so-called stiffness matrices, which are widely used today in the design of precision instruments. The development of applications for constant-force mechanisms [148,155,156] and high-precision positioners [157,158] is a characteristic of this cluster.

The research included in the green cluster goes beyond the kinematic simplifications of flexible-element deformations and seeks to address the nonlinear modeling of large deformations in flexible elements. One of the main weaknesses of the pseudo-rigid body method is that, when approximating large beam deflections using a rigid element with a single rotational pivot (1R), accuracy is lost at the end point and in the actual geometry of the flexible element. Seeking to represent these large displacements of flexible elements and joints as accurately as possible, our bibliometric analysis reveals three areas for improvement, although other classifications can be explored in [159]. On the one hand, advanced pseudo-rigid body models with multiple degrees of freedom have been proposed [160,161,162] to better adapt to the actual curvature of the flexible element. Other studies suggest solving the beam differential equations using elliptic integrals [163] or applying energy principles to obtain the well-known Beam Constraint Model [164,165], in its conventional or discrete versions, with several sub-elements in series [166].

Table 8. Interpretation of co-citation clusters, considering the oldest thematic fields.

Node	Cluster	Links	Citations	Interpretation	Reference
Bendsoe M.P. (2003)	1	37	116	Foundations of topology optimization and its application to continuum compliant mechanism synthesis	[144]
Sigmund O. (1997)	1	35	110		[121]
Pedersen C.BW. (2001)	1	29	40		[126]
Ling M. (2020)	2	30	39	Nonlinear modeling of large deflections in flexible links	[159]
Ma F. (2016)	2	28	23		[166]
Zhang A. (2013)	2	27	24		[163]
Lobontiu N. (2002)	3	42	165	Lumped compliance mechanism design methods based on pseudo-rigid-body modeling	[8]
Zhu B. (2020)	3	27	58		[167]
Howell L.L. (2013)	3	32	70		[145]
Howell L.L. (2001)	4	59	1237	Fundamental principles of compliant mechanisms and flexural hinges, and design fundamentals	[5]
Howell L.L. (2013)	4	40	244		[142]
Smith S.T. (2000)	4	26	109		[10]
Ananthasuresh G.K. (2003)	5	16	35	Synthesis and design of flexible mechanisms for MEMSs	[168]
Canfield S. (2000)	5	23	15		[169]
Murphy M.D. (1996)	5	14	24		[170]

Table 8. Interpretation of co-citation clusters, considering the oldest thematic fields.

Node	Cluster	Links	Citations	Interpretation	Reference
Bendsoe M.P. (2003)	1	37	116	Foundations of topology optimization and its application to continuum compliant mechanism synthesis	[144]
Sigmund O. (1997)	1	35	110		[121]
Pedersen C.BW. (2001)	1	29	40		[126]
Ling M. (2020)	2	30	39	Nonlinear modeling of large deflections in flexible links	[159]
Ma F. (2016)	2	28	23		[166]
Zhang A. (2013)	2	27	24		[163]
Lobontiu N. (2002)	3	42	165	Lumped compliance mechanism design methods based on pseudo-rigid-body modeling	[8]
Zhu B. (2020)	3	27	58		[167]
Howell L.L. (2013)	3	32	70		[145]
Howell L.L. (2001)	4	59	1237	Fundamental principles of compliant mechanisms and flexural hinges, and design fundamentals	[5]
Howell L.L. (2013)	4	40	244		[142]
Smith S.T. (2000)	4	26	109		[10]
Ananthasuresh G.K. (2003)	5	16	35	Synthesis and design of flexible mechanisms for MEMSs	[168]
Canfield S. (2000)	5	23	15		[169]
Murphy M.D. (1996)	5	14	24		[170]

The documents in the purple cluster reflect the early recognition of the inherent advantages of flexible mechanisms in MEMSs and show how the scientific community explored emerging design approaches for monolithic elements and systems at the micro-scale.

Finally, the central yellow cluster is characterized by works that synthesize the fundamental principles of flexible joints and mechanisms, as well as the basic principles of their analysis and design. For this same reason, their central location in the network indicates that they serve as a bridge between the different disciplines considered by the other groups.

Overall, the co-citation structure reveals a dual and complementary evolution of the field. One trajectory emphasizes analytical and kinematic modeling through localized compliance, while the other prioritizes computational optimization and continuum formulations. The interconnections among clusters indicate a progressive convergence of these approaches, driven by the need to address increasingly complex design challenges in precision engineering, micro-scale systems, and soft robotics.

3.2.3. Bibliographic Coupling

Applying the bibliographic coupling technique enables the identification of current and emerging thematic areas, as two documents are considered related when they cite a common set of references. This approach is particularly suitable for analyzing recent publications, which may not yet have accumulated a substantial number of citations but whose thematic proximity can be inferred from shared bibliographic foundations. Consequently, bibliographic coupling provides valuable insight into contemporary research directions and active knowledge fronts within the field.

Figure 14 presents the bibliographic coupling network generated using VOSviewer, with documents selected as the unit of analysis. The dataset was restricted to articles with a minimum of 20 citations indexed in Scopus, and the visualization displays the 50 strongest coupling relationships. The resulting network reveals four well-defined clusters, whose thematic interpretations and representative references are summarized in Table 9.

The bibliographic coupling and word co-occurrence analysis involve more recent articles, corresponding to a more advanced chapter of scientific development in the field of compliant mechanisms, characterized by well-consolidated applications and design and analysis techniques that contribute to extending the frontier of knowledge. For this reason, the results of these sections directly address RQ3 and RQ4. Overall, the bibliographic coupling analysis underscores the current diversification of compliant mechanism research, revealing a strong emphasis on computational design methods, multifunctional materials, and advanced numerical formulations. These trends reflect the field’s ongoing transition toward increasingly sophisticated and application-driven design paradigms.

The bibliographic coupling analysis can be regarded as the counterpart of the co-citation analysis, with a particular emphasis on recent publications and emerging research directions. As illustrated in Figure 14 and summarized in Table 9, although four distinct clusters are identified, the results clearly demonstrate the dominance of topology optimization as the preferred methodological framework for addressing a wide range of design and analysis challenges in compliant mechanisms. This prevalence highlights the flexibility of topology optimization in handling large deformations, complex boundary conditions, and multifunctional performance requirements.

Table 9 summarizes the thematic interpretation of the bibliographic coupling clusters and highlights the most representative publications within each group. Cluster 1 is associated with non-conventional geometries and shapes as optimal solutions in the design of compliant mechanisms and their elements. When a flexible mechanism is conceptually formulated using methodologies such as rigid-body substitution, a mechanism library, building blocks, or FACT, rigid-body architectures joined by flexible elements are typically considered. Originally, these elements had basic shapes, straight-line geometries, and constant cross-sections because, at the time, these considerations simplified the mathematics of the fundamental ideas that the original design methods sought to demonstrate. In contrast, the work of this cluster explores the implementation of curved elements [171] in the synthesis process. Interestingly, in work such as that of Liu [172], topological optimization is used not to design the flexible mechanism as a whole but rather to optimize a new multi-notched flexure hinge to achieve greater ranges of motion and precision. Review papers such as those by Bilancia [173], Ling [174], and Wang [148], which are present in this cluster, are a great reference for how nonstandard beam elements have been successfully used in different scenarios, especially in constant-force and torque devices.

Cluster 2 focuses on ensuring the manufacturability of flexible mechanisms designed based on topological optimization. Topological optimization remains central to this core research, but the focus is instead on developments that incorporate filters and constraints to control the size of material regions, ensuring the minimum size or volume necessary for manufacturability. The main methodological characteristic shared by the works in this cluster is the use of gradient-based methods, such as the level set method [175,176]. The works in this cluster demonstrate a concern not only for ensuring the manufacturability of prototypes at scales compatible with the capabilities of the machines used to manufacture them while also considering minimum scale aspects [177,178,179] but also for considering multi-material additive manufacturing of the mechanism from the outset [130,180].

Cluster 3 focuses entirely on a problem that has always been present in the results generated by topological optimization methods: the generation of facto hinges. This is a problem that has long been well known to the flexible mechanism research community [181,182], but what differentiates the work that makes up this cluster from that conducted in the past is the multi-criteria reformulation using methodologies such as BESO [183,184,185,186] or the level set method, where, in addition to hinge removal, multi-material manufacturing is used by employing local stress level constraints [187,188,189].

Cluster 4 reflects a transition from micro-scale applications of mechanisms fabricated via stereolithography, which typically involved small displacements relative to the mechanism’s size, to more conventional prototype scales manufactured using traditional cutting processes or additive manufacturing. This size characteristic naturally leads to large deformations that can no longer be assumed to be linear, as was done in early topological optimization results. This cluster is heavily reliant on evolutionary techniques, such as genetic algorithms, which are extremely useful for the structural optimization of flexible mechanisms with elements subjected to large nonlinear deformations, as well as for path generation.

Table 9. Interpretation of bibliographic coupling clusters, considering present thematic fields.

Label	Cluster	Links	Citations	Interpretation	Ref.
Lobontiu (2002)	1	47	841	Non-conventional geometries and shape optimization as optimal solutions in the design of compliant mechanisms and their elements	[8]
Wang (2018)	1	46	177		[148]
Hopkins (2010)	1	47	94		[136]
Liu (2017 b)	1	49	91		[172]
Lazarov (2016)	2	42	288	Topology optimization for manufacture of distributed compliant mechanisms	[190]
Gaynor (2014)	2	49	267		[180]
Wang (2005)	2	49	224		[175]
Luo (2007)	2	49	212		[176]
Chu (2018)	3	49	103	Topology optimization formulations for hinge-free designs of distributed compliant mechanisms	[187]
Zhu (2013)	3	49	66		[186]
Huang (2014)	3	49	60		[183]
Zhu (2012)	3	49	57		[185]
Yin (2003)	4	48	160	Nonlinear large deformations and path generation of distributed compliant mechanisms through topology optimization using evolutionary algorithms	[191]
Saxena (2005)	4	48	97		[192]
Rai (2007)	4	49	60		[193]
Saxena (2008)	4	49	54		[194]

Table 9. Interpretation of bibliographic coupling clusters, considering present thematic fields.

Label	Cluster	Links	Citations	Interpretation	Ref.
Lobontiu (2002)	1	47	841	Non-conventional geometries and shape optimization as optimal solutions in the design of compliant mechanisms and their elements	[8]
Wang (2018)	1	46	177		[148]
Hopkins (2010)	1	47	94		[136]
Liu (2017 b)	1	49	91		[172]
Lazarov (2016)	2	42	288	Topology optimization for manufacture of distributed compliant mechanisms	[190]
Gaynor (2014)	2	49	267		[180]
Wang (2005)	2	49	224		[175]
Luo (2007)	2	49	212		[176]
Chu (2018)	3	49	103	Topology optimization formulations for hinge-free designs of distributed compliant mechanisms	[187]
Zhu (2013)	3	49	66		[186]
Huang (2014)	3	49	60		[183]
Zhu (2012)	3	49	57		[185]
Yin (2003)	4	48	160	Nonlinear large deformations and path generation of distributed compliant mechanisms through topology optimization using evolutionary algorithms	[191]
Saxena (2005)	4	48	97		[192]
Rai (2007)	4	49	60		[193]
Saxena (2008)	4	49	54		[194]

3.2.4. Co-Word Analysis

Conducting a co-word analysis involves examining the co-occurrence of specific terms or keywords within a corpus of documents to identify semantic relationships among research topics, independent of citation frequency. This technique enables the detection of thematic clusters and the identification of emerging or declining research fronts within a field, depending on the temporal distribution of the analyzed documents. By uncovering conceptual linkages across studies, co-word analysis provides a content-oriented perspective that complements citation-based approaches.

Table 10. Top 75 trending topics (author keywords) identified using Bibliometrix.

Words	Occurrences	Words	Occurrences	Words	Occurrences
compliant mechanisms	2251	kinematics	62	neural network	36
topology optimization	457	pseudo-rigid-body model	61	optimal control	36
flexible manipulator	321	modeling	57	flexure mechanism	35
flexure hinges	318	parallel mechanism	57	sensitivity analysis	35
finite element analysis	218	robust control	57	parallel manipulator	34
flexible link	169	control	53	level set method	33
vibration control	158	trajectory tracking	53	motion control	33
mechanism design	152	constant-force mechanism	50	compliant mechanisms and robots	32
dynamic modeling	127	vibration	50	grasping	32
optimization	119	variable stiffness	49	gripper	32
piezoelectric actuator	117	flexible link manipulator	47	origami	32
soft robotics	93	force control	47	compliant parallel mechanism	31
compliant joint mechanisms	90	manipulator	44	compliant gripper	30
dynamics	89	simulation	44	multi-objective optimization	30
additive manufacturing	80	flexible	43	robotics	30
compliance	80	large deflection	42	compliant	29
adaptive control	78	singular perturbation	42	design	29
vibration suppression	76	flexible-link manipulator	41	mechanical design	29
mems	74	stiffness	41	stability	29
flexible arm	73	active vibration control	39	bistable mechanisms	28
flexible robot	71	mechanism synthesis	37	dynamic analysis	28
microgripper	70	robot	37	screw theory	28
robot design	69	shape optimization	37	system identification	28
sliding mode control	69	design optimization	36	fuzzy control	27
genetic algorithm	68	flexible joint	36	input shaping	27

presents the 75 most frequent author keywords identified using Bibliometrix. The resulting keyword set reveals both the core and peripheral subfields that characterize research on compliant mechanisms. From these data, several principal thematic axes can be identified.

The first axis corresponds to design and modeling methodologies, represented by keywords such as finite element method, topology optimization, structural optimization, and shape optimization. The second axis is associated with dynamic behavior and control aspects, including flexible manipulators, vibration analysis, motion control, controllers, and adaptive control systems. The third axis reflects applications in robotics and precision engineering, highlighted by terms such as grippers, end effectors, robotic arms, and parallel mechanisms. The fourth thematic axis is related to actuation technologies and mechanical elements, including piezoelectric actuators, the pseudo-rigid-body model, and flexure hinges. Finally, the fifth axis encompasses computational and numerical techniques, evidenced by keywords such as genetic algorithms, numerical methods, and finite element analysis.

The prominence of these thematic axes highlights the dual nature of compliant mechanism research. On the one hand, it remains a theoretically grounded discipline focused on mechanical modeling, optimization, and analytical rigor. On the other hand, it is strongly application-driven, with increasing emphasis on robotics, microelectromechanical systems, and precision devices. Furthermore, the presence of emerging keywords such as additive manufacturing, biomimetics, and soft robotics (although with lower frequencies) indicates a progressive shift toward novel design paradigms involving smart materials, additive manufacturing technologies, and biologically inspired concepts.

In addition, Figure 15 illustrates the co-occurrence network of author keywords generated using VOSviewer, considering a minimum threshold of seven occurrences.

The analysis reveals the presence of 13 distinct keyword clusters, formed based on the frequent co-occurrence of terms within the body of the literature, independently of their temporal distribution. These clusters represent coherent thematic groupings within compliant mechanism research. VOSviewer automatically assigns a unique color to each cluster for visualization purposes: red (cluster 1), green (cluster 2), blue (cluster 3), yellow (cluster 4), purple (cluster 5), cyan (cluster 6), orange (cluster 7), brown (cluster 8), pink (cluster 9), light orange (cluster 10), light green (cluster 11), light purple (cluster 12), and light yellow (cluster 13). The thematic interpretation of these clusters is presented and discussed in detail in

Table 11. Interpretation of co-word clusters, considering present and future thematic fields.

Label	Cluster	Links	Avg. Pub. Year	Avg. Citations	Interpretation	Ref.
flexible manipulator	1	66	2011.7	21.9	Vibration Control and Automation in Flexible Manipulators	[195]
flexible link		61	2013.5	15.8		[196]
vibration control		58	2013	26.1		[197]
sliding mode control		45	2015.3	17.9		[198]
adaptive control		44	2014.7	20.5		[199]
dynamics	2	65	2016.3	16.6	Advanced Compliant Mechanisms and Bio-Inspired Robotics	[200]
kinematics		48	2017.8	18.7		[201]
compliant mechanisms and robots		30	2024.8	3		[202]
grasping		25	2020.7	27.4		[203]
continuum robot		21	2022.2	6.9		[204]
dynamic modeling	3	56	2015.5	16.3	Evolutionary Optimization and Dynamic Modeling of Flexible Systems	[205]
genetic algorithm		33	2013.9	21.8		[206]
flexible link manipulator		28	2014.8	12.5		[207]
particle swarm optimization		27	2017	23.3		[208]
fuzzy control		23	2008.8	26.3		[209]
optimization	4	61	2016.3	11.6	Fundamentals of Mechanical Design and Structural Compliance	[210]
compliance		39	2017	27.9		[211]
design		30	2016.2	17.7		[212]
dynamic analysis		22	2013.4	9.6		[213]
compliant		21	2016.6	12.7		[214]
topology optimization	5	51	2016.6	39.6	Advanced Structural and Topology Optimization	[215]
additive manufacturing		25	2021.6	17.2		[216]
design optimization		23	2019.7	14.3		[217]
shape optimization		20	2013.9	44.3		[218]
sensitivity analysis		18	2013.5	34.8		[219]
mems	6	29	2013.5	25.5	Constant-Force Mechanisms, Variable Stiffness, and Bistable Devices	[220]
variable stiffness		29	2020.7	12.4		[221]
constant-force mechanism		22	2020	30.9		[222]
bistable mechanisms		20	2017.3	23.8		[223]
gripper		19	2019.3	25.8		[224]
flexure hinges	7	70	2017.1	23	Micro-Flexure Hinges and Nanopositioning Guidance Systems	[158]
piezoelectric actuator		36	2017.3	26.8		[225]
flexure mechanism		19	2018.7	13.6		[226]
mechanical design		18	2018.7	26.7		[227]
large stroke		14	2021.1	11.1		[228]
modeling	8	48	2015.4	13.4	Classical Modeling of Robotic Frameworks and Simulation	[229]
control		41	2013	18.1		[230]
manipulator		36	2013.8	15.8		[231]
robot		31	2007.1	11.2		[232]
vibration		31	2011.7	11.9		[233]
simulation	9	34	2012.5	10.7	Closed-Loop Position Control and Kinematic Simulation	[234]
flexible		29	2015.4	10.8		[235]
robots		28	2014.7	22.7		[236]
position control		21	2011.1	13.3		[237]
lqr		17	2016.7	14.6		[238]
mechanism design	10	49	2020.8	17.9	Soft Robotics and Kinematic Mechanism Synthesis	[239]
robot design		43	2016.7	12.4		[240]
soft robotics		43	2022.2	21.9		[241]
mechanism synthesis		25	2020.1	12.9		[242]
bio-inspired design		20	2021.7	13.9		[243]
parallel mechanism	11	32	2014.9	15	Parallel Mechanisms and Structural Stiffness Profiling	[244]
stiffness		27	2015.9	20.5		[245]
parallel manipulator		25	2015.1	21.2		[246]
compliant joint		19	2018.4	11.7		[247]
finite element		17	2012.9	23.8		[248]
compliant mechanisms	12	121	2017.9	20.1	Compliant Mechanism Theory and Pseudo-Rigid-Body Modeling	[249]
finite element analysis		70	2015	12		[250]
large deflection		21	2016	15.1		[251]
pseudo-rigid-body model		19	2015.9	25.3		[252]
geometric nonlinearity		10	2018.1	20.3		[253]
microgripper	13	26	2016.9	25.7	Micromanipulation, Workspace Characterization, and Analytical Approximations	[254]
micromanipulation		18	2016.4	16.6		[255]
workspace		16	2017.2	23.5		[256]
compliant parallel mechanism		15	2018.7	11.3		[257]
assumed mode method		11	2015.2	13.3		[258]

.

The keyword clusters identified through co-word analysis are summarized in Table 11, which provides a concise interpretation of each cluster’s thematic focus, along with representative references and the average publication year of the associated keywords. Taken as a whole, these clusters delineate the conceptual structure of the compliant mechanism research landscape, offering a comprehensive overview of both its current state and emerging directions. This analytical approach enables the identification of present and prospective thematic domains. Clusters with higher average publication years (closer to 2025) generally correspond to recently consolidated or rapidly expanding research fronts, whereas clusters with lower average publication years tend to represent established and foundational areas of the discipline. In this regard, Table 11 not only maps the intellectual organization of the field but also serves as a practical guide for researchers seeking promising avenues for innovation. Moreover, these results confirm that compliant mechanism research is transitioning from a predominantly theoretical focus—centered on kinematics and mechanical modeling—toward an increasingly interdisciplinary landscape that integrates advanced materials, smart actuation, advanced control robotics, and digital fabrication technologies. The thematic clusters characterized by more recent average publication years, in particular, highlight the research frontiers that are actively shaping the future of compliant mechanism design.

The interpretation of each group on its own indicates the general theme of the works containing those keywords. However, the authors may be overlooking mathematical tools such as Lyapunov stability and candidate functions, as well as neural networks, used for nonlinear identification systems [198,199,206,230,233].

Studying the references in Table 8, Table 9 and Table 10 reveals various applications of compatible mechanisms to aerospace problems [66,181,196,209,229,234], using a multidisciplinary approach encompassing basic and specialized areas such as compatible mechanisms, robotics, soft robotics, advanced control theory, and vibration suppression. Interestingly, works such as [259] are considered in the initial dataset downloaded from Scopus due to the inclusion of the term “compliant mechanisms” in their keywords and abstracts. However, their subsequent development [260] does not appear in the downloaded dataset. It is undeniable that the principles and concepts of compliant mechanisms are present in these works; however, as a reflection, we could take this as an example to show that the theoretical development of flexible mechanisms has reached a level of maturity such that works demonstrating their application or a high degree of specialization will assume that the reading public understands the general context of compliant mechanisms in order to concentrate their efforts on the main problem addressed by the investigation in question.

It is remarkable how the pseudo-rigid-body model has transcended its seminal origins to remain a relevant subject of study in current trends. Clearly, it is not the “same” pseudo-rigid-body model proposed by Midha and Howell over 35 years ago; it is a model that has been refined over time and is increasingly being applied across diverse fields and scientific disciplines, where it can be combined to provide novel solutions to design problems. The distribution of author keyword clusters (Figure 16) further supplements the historical conceptual turning points outlined in the Introduction. The chronological transition of terms proves that the discipline did not expand randomly; rather, it migrated structurally from foundational “flexure hinge” modeling toward advanced “topology optimization” and “soft robotics” clusters. The literature metrics supplement traditional historical tracking by demonstrating that the conceptual framework established by Midha and Howell regarding restricted flexible links did not merely add citations but also structurally bifurcated the field’s taxonomy—shifting the thematic momentum from classical kinematic synthesis to advanced distributed compliance, smart actuation, and localized microelectromechanical systems (MEMSs).

Figure 16 presents the same keyword co-occurrence network shown in Figure 15, using the overlay visualization option to represent the average publication year of each keyword within a color scale ranging from 2005 to 2025. This visualization enables a temporal interpretation of keyword evolution, where node colors indicate the relative maturity or novelty of research topics within the compliant mechanism domain.

Keywords shown in green correspond to well-established and foundational research areas, such as kinematic modeling, finite element analysis, flexure hinges, and vibration analysis, which have historically supported the theoretical development of compliant mechanism design. In contrast, keywords highlighted in red and orange represent more recent and rapidly growing research themes, including compliant mechanisms in robotics, additive manufacturing, bio-inspired design, soft robotic systems, displacement amplifiers, constant-force mechanisms, and flexible link manipulators.

The prominence of these emerging keywords reflects a clear shift in research focus from predominantly analytical and kinematic formulations toward application-driven and interdisciplinary studies. In particular, the increasing occurrence of terms related to digital fabrication, smart materials, and multifunctional actuation indicates that compliant mechanism research is increasingly influenced by advances in manufacturing technologies and material engineering. Furthermore, the convergence of keywords associated with robotics, control, precision engineering, and adaptive systems highlights the growing relevance of compliant mechanisms in areas requiring high accuracy, lightweight structures, and integrated compliance control.

Overall, the overlay visualization confirms that the field of compliant mechanisms is undergoing a progressive transformation, in which classical modeling approaches coexist with emerging paradigms centered on smart actuation, additive manufacturing, and bio-inspired design. These trends suggest that future research will increasingly emphasize system-level integration, multifunctionality, and adaptability, positioning compliant mechanisms as key enablers in next-generation mechanical and robotic systems.

Clustering derived from co-word analysis reveals the large-scale conceptual structure of the compliant mechanism research field. When combined with overlay visualization, this approach enables the identification of the most recent and influential concepts, as well as their interconnections, thereby facilitating the exploration of emerging relationships that can inspire novel research directions.

Complementing the interpretations drawn from Figure 15 and Figure 16 and Table 11, Figure 17 presents the trending topics identified using the Bibliometrix package. This analysis was performed using a minimum keyword frequency of five and a maximum of three words per term per year, allowing for a clear depiction of topic dynamics over time.

The co-word analysis summarized in Table 10 highlights the research themes that were most extensively studied throughout the entire period considered in this review, reflecting the foundational knowledge base of the field. In contrast, Figure 17 illustrates the evolution of research topics over the past 25 years and emphasizes those that have only recently emerged or gained significant momentum. Notably, these include compliant mechanisms applied to robotics and soft robotics, bio-inspired design, additive manufacturing, piezoelectric actuators, microgrippers, and vibration isolators. These emerging and rapidly expanding topics represent promising research frontiers in which future efforts may be most impactful. In particular, they constitute fertile ground for conducting systematic and focused review studies, which can help identify knowledge gaps, refine existing methodologies, and formulate new research questions that drive further theoretical and technological advancements in compliant mechanism design.

3.2.5. Co-Authorship Analysis

Co-authorship analysis enables the examination of collaborative relationships among authors, institutions, and countries within the compliant mechanism research community. This approach provides insight into the structure of scientific collaboration by identifying well-established research groups, highly active international collaboration networks, and potential cases of institutional or geographic isolation that may merit further investigation. Such patterns are valuable for understanding how knowledge is generated and disseminated across the field and may inform future strategies aimed at strengthening collaboration and knowledge exchange.

Figure 18 illustrates the co-authorship network constructed for authors with a minimum of seven publications and at least 25 citations. This threshold was selected to emphasize the collaboration patterns of highly productive and influential researchers, as well as the research groups to which they are affiliated, thereby offering a focused representation of the core collaborative structure of the field.

Building on the previous analysis, Figure 19 presents the international collaboration network among countries, including only those with at least five publications on compliant mechanisms, irrespective of citation counts. This threshold was selected to capture meaningful patterns of cross-country collaboration while ensuring adequate representation across the dataset. During the data preprocessing stage, careful metadata cleaning was required to ensure the accuracy of the collaboration network. In particular, the initial network erroneously included “IEEE” as a country, which was identified and removed using conditional filtering of the affiliation field in Excel. Additionally, 54 instances in which the United States appeared under the abbreviation “USA” were standardized to ensure consistency in country naming.

The resulting co-authorship analysis provides a clear visualization of international collaboration patterns, revealing a strong level of cooperation among leading countries and research groups. This widespread interconnectedness suggests that research on compliant mechanisms is not constrained by disciplinary or geographic barriers but rather benefits from an open and collaborative scientific environment that supports the global development of the field.

4. Trends and Perspectives

The temporal coverage of the present bibliometric analysis enables a robust identification of the principal trends, structural transformations, and prospective research directions within the field of compliant mechanisms. The accumulated evidence indicates that this research area has reached a high level of scientific maturity. While foundational studies continue to evolve through refined theoretical formulations and advanced synthesis methodologies, contemporary research clearly reflects a paradigm shift from predominantly theoretical investigations toward application-oriented and technology-driven developments, strongly supported by interdisciplinary integration.

The sustained growth in scientific productivity over the analyzed period confirms that compliant mechanisms remain a highly dynamic and relevant topic within the global research community. Rather than approaching conceptual saturation, the field continues to diversify through the incorporation of novel materials, innovative actuation strategies, and increasingly sophisticated computational and optimization-based design frameworks. This diversification has expanded both the functional scope and the technological relevance of compliant mechanisms across multiple engineering domains.

Historically, as evidenced by the co-citation clusters presented in Table 8, early research efforts largely focused on establishing the feasibility of synthesizing mechanisms with flexible elements and accurately predicting their kinematic and static behaviors. These studies laid the theoretical foundations necessary for understanding compliance, motion transmission, and force–displacement relationships. In contrast, current research trends emphasize the deployment of compliant mechanisms in practical and high-performance applications, leveraging their inherent advantages—such as lightweight construction, reduced assembly complexity, monolithic fabrication, and high precision—to meet discipline-specific functional requirements. Prominent application hotspots include soft and parallel robotics, ultra-high-precision engineering, biomedical and surgical devices, microelectromechanical systems (MEMSs), and morphing or multistable structures.

In parallel, the strong convergence between compliant mechanism design and additive manufacturing technologies has emerged as one of the most rapidly expanding research frontiers. Advances in multi-material 3D and 4D printing have enabled the realization of complex compliant architectures that were previously impractical or impossible to fabricate using conventional manufacturing techniques. These include bio-inspired structures, origami- and kirigami-based mechanisms, and cellular or mechanical metamaterials with tunable stiffness, anisotropic behavior, or programmable deformation paths. As additive manufacturing technologies continue to mature, this research avenue is expected to facilitate the development of adaptive, self-reconfigurable, and functionally graded compliant systems, further blurring the traditional boundaries between structural design and material functionality.

Another rapidly growing trend involves the integration of piezoelectric and other smart materials into compliant mechanisms. Piezoelectric actuators, in particular, have enabled the development of micro- and nano-positioning platforms, precision grippers, vibration control devices, and stepping actuators operating at the limits of mechanical resolution. Similarly, shape-memory alloys (SMAs) and electroactive polymers are increasingly incorporated as active or deformable elements, providing self-actuation and multifunctionality. However, these materials also introduce strong nonlinearities, hysteresis effects, and coupled field interactions, which continue to motivate advances in modeling, control strategies, and experimental validation.

At the same time, the need for accurate dynamic modeling and advanced control strategies has intensified. As compliant mechanisms are increasingly deployed in high-speed, high-frequency, and small-scale applications, understanding their dynamic behavior, vibration characteristics, and energy dissipation mechanisms has become critical for ensuring precision, reliability, and long-term performance. This growing emphasis on system dynamics reflects a broader transition from quasi-static design approaches toward fully dynamic and multiphysics optimization frameworks, where coupled mechanical–electrical–thermal models are becoming standard practice.

In summary, the field of compliant mechanisms is currently at a pivotal stage of evolution. It has progressed beyond the establishment of its theoretical foundations and is now actively exploring high-impact applications and hybrid technologies that integrate mechanics, materials science, control engineering, and advanced manufacturing. The findings of this bibliometric analysis suggest that future research efforts will increasingly focus on:

(a)

The integration of intelligent, multifunctional, and self-sensing materials to enable adaptive and autonomous compliant systems;

(b)

The application of digital design methodologies, machine learning, and artificial intelligence for topology synthesis, motion optimization, and performance prediction;

(c)

The expansion of additive and 4D manufacturing techniques for scalable, reconfigurable, and functionally graded compliant mechanisms;

(d)

The exploration of emerging application domains, including biomechanics, aerospace morphing structures, precision instrumentation, and soft robotic systems.

Ultimately, compliant mechanisms represent a vibrant and rapidly evolving research domain, well positioned to address complex multidisciplinary challenges. Their increasing convergence with smart materials, digital design tools, and advanced fabrication technologies promises a transformative impact on both fundamental research and the practical realization of next-generation mechanical and robotic systems.

5. Conclusions

This study successfully develops a systematic review analysis supported by a bibliometric analysis of design methodologies for compliant mechanisms, with the integrated bibliometric analysis guided by the PRISMA methodology. The adaptation of PRISMA to a bibliometric context introduced a procedure equivalent to the risk of bias assessment traditionally used in systematic reviews, ensuring transparency and reproducibility in data collection and analysis. This methodological contribution, illustrated conceptually in Figure 3, represents a significant step forward in applying systematic review standards to engineering bibliometrics. The following conclusions summarize the main findings and methodological contributions derived from this work:

➢

The application of PRISMA principles led to the design of an optimized preprocessing strategy that identified and corrected metadata inconsistencies and duplicate records. This approach yielded a refined dataset of 10,425 documents retrieved from Scopus, thereby supporting the reliability of the results. The methodology also proved useful for identifying innovative studies that illustrate the versatility of compliant mechanisms, such as the simulation of DNA origami nanoactuators and nanostructures inspired by flexible mechanisms.

➢

The cumulative publication trend from 1966 to 2025 confirms that research on compliant mechanisms remains a vigorous and expanding field, with an average annual growth rate of 12.02%. The main sources of dissemination are Proceedings of the ASME Design Engineering Technical Conference (486 publications), Mechanism and Machine Theory (361), and Journal of Mechanisms and Robotics (272), reflecting a strong preference for specialized engineering venues.

➢

From a geographical perspective, China and the United States dominate global productivity, contributing 10,455 and 6207 publications, respectively. They are followed by India, Germany, Japan, Italy, and Canada, each with over 1000 publications. This pattern aligns with the nationality of the two most prolific authors in the field: Larry Howell (USA, 173 publications) and Xianmin Zhang (China, 164 publications), who have shaped the development of analytical and optimization-based design approaches for compliant mechanisms.

➢

The citation and reference analyses revealed the foundational works that define the discipline’s conceptual core. Seminal contributions by Sigmund [121], Howell [1,5], and Frecker [122] remain the most influential, alongside key monographs by Smith [10] and Lobontiu [8]. Early works, such as those by Burns [26], continue to be recognized for establishing the theoretical basis of flexible link synthesis. The inclusion of such classical and gray literature demonstrates the methodological robustness of this review and its capacity to capture the field’s historical continuity.

➢

Co-citation and bibliographic coupling analyses revealed five and four original thematic cores, respectively, that structure the intellectual evolution of compliant mechanism design: the foundations of topology optimization and its application to continuum compliant mechanism synthesis, nonlinear modeling of large deflections in flexible links, lumped compliance mechanism design methods based on pseudo-rigid-body modeling, fundamental principles of compliant mechanisms and flexural hinges and design fundamentals, and synthesis and design of flexible mechanisms for MEMSs.

➢

More recent analyses show the emergence of application-oriented subfields integrating topological optimization, multimaterial systems, and metamaterials. Current research emphasizes parallel manipulators, hinge elimination, additive manufacturing, and finite element modeling, reflecting a steady evolution toward computational and multifunctional design paradigms.

➢

The co-word analysis identified 13 keyword clusters representing consolidated and emerging research fronts. Mature areas (average publication year ≤ 2015) include vibration control, parallel mechanism dynamics, flexure-based mechanisms, and energy harvesting. Meanwhile, emerging areas (average year ≥ 2020) highlight soft robotics, microgrippers, 4D-printed compliant mechanisms, transfer matrix methods, microactuators, and variable-stiffness structures. These topics exemplify the ongoing convergence between compliant mechanisms, smart materials, and digital fabrication technologies.

➢

The bibliometric evidence thus confirms that the field has not reached saturation but continues to diversify and evolve by integrating intelligent materials, additive manufacturing, and advanced computational techniques. The methodology developed in this study can be replicated or adapted to shorter time spans or to the emerging keywords identified here, allowing researchers to track new research frontiers as they grow.

➢

Finally, compliant mechanisms represent a mature yet rapidly evolving field characterized by methodological innovation, theoretical depth, and broad applicability. The combination of flexible structural elements, distributed compliance, and emerging smart materials redefines the design paradigm of mechanical and mechatronic systems. The outcomes of this study provide both a conceptual map for understanding the field’s past and present and a strategic framework for guiding future investigations toward intelligent, adaptive, and sustainable mechanical design.