Flexible Joint Research Articles

Analytical modeling and computational optimization for a 1-DOF compliant mechanism

The first natural frequency directly affects the operational conditions of compliant mechanisms in precision engineering systems. To address this challenge, a computational method based on the surface response method is proposed to optimize the frequency of a new one degree of freedom (DOF) compliant mechanism. Initially, a 1-DOF 3D model of a compliant mechanism is built. The dynamic equation and the frequency response are formulated via equivalent stiffness and the Lagrange method. Subsequently, a series of numerical simulations are conducted to find the fundamental frequency of the mechanism. The initial dimensions of the flexible joints are determined, and the initial frequency is analyzed by using finite element analysis. Next, the flexible joints in the designed mechanism are optimized by a variant of the genetic algorithm. The optimized dimensions of the mechanism are found with the thickness of the circular joint of 1.10 mm, the thickness of the leaf joint of 1.19 mm, and the length of the leaf joint of 54.50 mm. The optimized result showed that there is a significant improvement in the frequency of the mechanism, increasing from an initial design of 53.218 Hz to an optimal design of 75.927 Hz with an improvement of 42.6%. This study provides important reference materials for future research on compliant mechanisms.

LJF of thin-walled gapped uni-planar K-type tubular steel joints with collar under brace balanced axial loads

This paper investigates the efficacy of the collar plate in improving the Local Joint Flexibility (LJF) of thin-walled circular hollow section (CHS) K-joints under axial loads. Initially, a finite element model (FEM) was created and verified against data from 14 available experimental tests. Subsequently, an extensive series of 136 unreinforced and reinforced K-joints was simulated to analyze the influence of parameters such as collar plate dimensions (η and λ), brace angle (θ), and joint geometry (β, ξ, and γ) on the LJF factor (fLJF) and the fLJF ratio (χ). The FEMs incorporated plate-to-member contact and weld modeling. Findings indicate that the utilization of the plate reduces the fLJF by 73 %. Additionally, it was observed that plate length significantly affects the fLJF compared to plate thickness. While the impact of γ, θ, and ξ on χ is marginal, β, η and λ emerge as a noteworthy determinant of χ. Despite the pivotal role of fLJF in joint behavior, there exists a gap in studies and equations specifically addressing fLJF in collar-plate reinforced K-joints under axial loads. To address this gap, leveraging the FE results, a design equation is proposed. This equation is subsequently validated with high coefficients of determination and standards set by the UK Department of Energy.

Transforming rehabilitation: A narrative review on reversible knee flexor contracture assistive devices

Abstract: Joint contracture, marked by restricted joint movement due to connective tissue and muscle shortening, is a common complication in chronic musculoskeletal conditions such as osteoarthritis, osteonecrosis, rheumatoid arthritis, healed septic joint, and postsurgical complications. This limitation adversely impacts joint mobility and flexibility, increasing the likelihood of physical constraints. Contractures elevate the risk of impaired self-care, limited physical mobility, and hindered social activities, emphasizing the critical need to manage such contractures. The study aims to find the most appropriate, effective, user-friendly mechanical device to treat reversible knee flexor contracture. Method of a literature review was conducted utilizing PubMed, Google Scholar, and the physiotherapy evidence database (PEDro) up to April 2023. The inclusion criteria comprised studies related to flexion contracture (FC), written in English languages were included. The literature searched using the terms “flexion contracture, hamstring contracture, knee flexor contracture, burn contracture, knee joint hypomobility, and devices for flexion contracture.” Result after applying the selection criteria, the initial screening of literature gives 35,400 results on Google Scholar, PubMed, and PEDro. Subsequently, 128 articles underwent screening based on abstract and full-text availability in the English language. Following this, seven articles were selected and thoroughly reviewed, which included randomized control trials, systematic reviews, and exploratory studies. The study concluded the use of conventional physiotherapy interventions, coupled with assistive devices, diminishes the burden on physiotherapists and provides effective improvements to patients with FCs.

Effects of Fatigue on Ankle Flexor Activity and Ground Reaction Forces in Elite Table Tennis Players.

Fatigue specifically affects the force production capacity of the working muscle, leading to a decline in athletes' performance. This study investigated the impact of fatigue on ankle flexor muscle activity and ground reaction forces (GRFs) in elite table tennis players, with a focus on the implications for performance and injury risk. Twelve elite male table tennis athletes participated in this study, undergoing a fatigue protocol that simulated intense gameplay conditions. Muscle activity of the soleus (SOL) and gastrocnemius lateralis (GL) muscles, heel height, and GRFs were measured using a combination of wireless electromyography (EMG), motion capture, and force plate systems. Results showed a significant decrease in muscle activity in both legs post-fatigue, with a more pronounced decline in the right leg. This decrease in muscle activity negatively affected ankle joint flexibility, limiting heel lift-off. Interestingly, the maximal anteroposterior GRF generated by the left leg increased in the post-fatigue phase, suggesting the use of compensatory strategies to maintain balance and performance. These findings underscore the importance of managing fatigue, addressing muscle imbalances, and improving ankle flexibility and strength to optimize performance and reduce the risk of injuries.

An Assessment of the Impact of Protective Lifeline Safety Systems on Formwork Systems

Work related to the installation of formwork and the pouring of concrete involves the need to move on the formwork walls. There is a high risk of falling off the wall and falling during work. It is therefore necessary to use systems that allow for safe work at heights, but also the safety of the formwork systems themselves. This article presents the results of tests under real conditions of a person falling from the formwork. Two safety systems were tested: one based on a pole–rope system with a flexible joint and one based on climbing arrester hooks. The test results showed that the additional forces occurring in the system do not exceed 6.4 kN and that the stresses reach the highest value only at the site of installation of the system and do not exceed the yield strength of the steel used in the formwork structure. The results obtained indicate that during a fall, the fall energy is absorbed so much that there is no damage to the formwork elements, and the highest effort is observed for the formwork arrester hook and reaches 87% of the structure effort. In the case of systems with flexible posts, the maximum load values do not exceed 30% of the structure. The presented results may be useful in designing and planning assembly works and using formwork systems in concrete pouring conditions.

Experimental study on mechanical behavior and countermeasures of mountain tunnels under strike-slip fault movement

In the seismic mountainous regions such as western China, it is usuallly inevitable to construct tunnels near active fault zones. Those fault-crossing tunnel structures can be extremely vulnerable during earthquakes. Extensive experimental studies have been conducted on the response of continuous mountain tunnels under reverse and normal fault movements, limited experimental investigations are available in the literatures on mountain tunnels with special structural measures crossing strike-slip faults. In this study, a new experimental facility for simulating the movement of strike-slip fault was developed, accounting for the spatial deformation characteristics of large active fault zones. Two groups of sandbox experiment were performed on the scaled tunnel models to investigate the evolution of ground deformation and surface rupture subjected to strike-slip fault motion and its impact on a water conveyance tunnel. The nonlinear response and damage mechanism of continuous tunnels and tunnels incorporated with specially designed articulated system were examined. The test results show that most of slip between stationary block and moving block occurred within the fault core, and significant surface ruptures are observed along the fault strike direction at the fault damage zone. The continuous tunnel undergoes significant shrinkage deformation and diagonal-shear failure near the slip surface and resulted in localized collapse of tunnel lining. The segments of articulated system tunnel suffer a significant horizontal deflection of about 5°, which results in opening and misalignment at the flexible joint. The width of the damaged zone of the articulated system tunnel is about 0.44 to 0.57 times that of the continuous tunnel. Compared to continuous tunnels, the articulated design significantly reduces the axial strain response of the tunnel lining, but increases the circumferential tensile strain at the tunnel crown and invert. It is concluded that articulated design provides an effective measure to reduce the extent of damage in mountain tunnel.

Dynamic modelling and a model-based control strategy of joint servo system for power transmission line inspection robot

Abstract The time-varying load inertia is one of the important dynamic characteristics of the power transmission line inspection robots (PTLIRs) in joint space, which can heavily affect the stability of the robot in the inspection process. In this paper, with considering the time-varying characteristics of load inertia, a method for modelling and control of the flexible joint servo system of the PTLIR is studied. First, the control system of the PTLIR is established based on the two inertia system. In order to calculate the load inertia, a dynamic model of the inspection robot is established. Then, the model-based notch filter is designed to improve dynamic performance of the robot, and suppress the vibration amplitude change which caused by the time-varying load inertia. Finally, the comparison of the control performance between the traditional control strategy and the model-based control strategy with notch filter are compared. The simulation results show that the paper can provide an effective tool for modelling and improving the stability of the PTLIR.

Multi-Behavior Recommendation with Personalized Directed Acyclic Behavior Graphs

A well-developed recommendation system can not only leverage multi-typed interactions (such as page view, add-to-cart, and purchase) to better identify user preferences, but also demonstrate high performance, low complexity, and strong interpretability. However, many existing solutions for multi-behavior recommendation fall short of intuitive modeling of real-world scenarios, leading to overly complex models with massive parameters and cumbersome components. In particular, they share two critical limitations: (1) Some pioneering models are built upon the strict assumption of cascade effects across behaviors, which contradicts multifarious behavior paths in practical applications. (2) Existing approaches fail to explicitly capture the unique idiosyncrasies of users and even neglect the inherent nature of items involved in the multi-behavior interactions. To this end, we propose a novel Directed Acyclic Graph Convolutional Network (DA-GCN) for the multi-behavior recommendation task. Specifically, we pinpoint the partial order relations within the monotonic behavior chain and extend it to personalized directed acyclic behavior graphs to exploit behavior dependencies. Then, a GCN-based directed edge encoder is employed to distill rich collaborative signals embodied by each directed edge. In light of the information flows over the directed acyclic structure, we propose an attentive aggregation module to gather messages from all potential antecedent behaviors, representing distinct perspectives to understand the terminated behavior. Thus, we obtain comprehensive representations for the follow-up behavior through learnable distributions over its preceding behaviors, explicitly reflecting personalized interactive patterns of users and underlying properties of items simultaneously. Finally, we design a customized multi-task learning objective for flexible joint optimization. Extensive experiments on public benchmarking datasets fully demonstrate the superiority of DA-GCN with significant performance improvement and computational efficiency over a wide range of state-of-the-art methods. Our code is available at https://github.com/xizhu1022/DA-GCN .

Understanding medicine related adverse events in people with multiple long-term conditions and polypharmacy

Background and ObjectivesPolypharmacy is increasingly common in people living with multiple long-term conditions (MLTC), but clinical evidence often does not consider medicine interactions beyond two or three treatments at a time. The aim of this research is to develop statistical models to better understand interactions between treatments leading to adverse events for people living with MLTC. ApproachWe will develop models of risk of adverse events in people with MLTC and polypharmacy using data in the Secure Anonymised Information Linkage (SAIL) Databank - a trusted research environment with linked electronic health record data at the individual-level for the population of Wales, UK. Taking a flexible Bayesian joint modelling approach, where the posterior risk is the product of several sub-models, will enable the characterisation of this complex system in a way that is interpretable for healthcare decision-makers, leading to improved understanding of risk factors for adverse events. ResultsThe Welsh Multimorbidity e-Cohort (WMC) includes 2.9 million people alive and living in Wales on 1st Jan 2000 with follow-up for 20 years. Individuals in the cohort have 932,552,061 medication items prescribed and 277,279 coded adverse events recorded in primary care data. Modelling approaches will focus on feasibility of fitting models to population-level datasets, and the interpretability of the model results for healthcare decision-making. Conclusions and ImplicationsResults have the potential to inform healthcare policy and practice for the management of people living with MLTC and polypharmacy, thus improving patient outcomes and making the best use of limited healthcare resources.

Exercise for patients with arthritis

Background: Osteoarthritis and rheumatoid arthritis are major causes of pain and disability globally, and exercise is known to be effective at alleviating symptoms and improving physical function.Current Concepts: Knee osteoarthritis is a common degenerative disease, and comprehensive clinical practice guidelines have been established. Most studies strongly recommend appropriate weight management combined with exercise therapy. Aerobic, strength, resistance, isometric, neuromuscular, and balance training have been used. However, the current evidence is insufficient to recommend specific prescriptions regarding the exercise type and dose. Many guidelines group knee and hip osteoarthritis together for exercise recommendations. Evidence supporting the use of exercise for hand osteoarthritis is relatively scarce, and reports on its adverse effects are rare. Weight loss is not recommended for hand osteoarthritis, because its effectiveness has not yet been established. For patients with polyarticular osteoarthritis, exercise is recommended as a core treatment, with systematic land-based exercises being particularly advised. For rheumatoid arthritis, both general exercises and hydrotherapy are recommended to improve joint flexibility, muscle strength, and functional outcomes.Discussion and Conclusion: Exercise is a powerful treatment tool for various types of arthritis. However, further research is required to develop more specific exercise intervention protocols.

Flexible joint model for time-to-event and non-Gaussian longitudinal outcomes.

In medical studies, repeated measurements of biomarkers and time-to-event data are often collected during the follow-up period. To assess the association between these two outcomes, joint models are frequently considered. The most common approach uses a linear mixed model for the longitudinal part and a proportional hazard model for the survival part. The latter assumes a linear relationship between the survival covariates and the log hazard. In this work, we propose an extension allowing the inclusion of nonlinear covariate effects in the survival model using Bayesian penalized B-splines. Our model is valid for non-Gaussian longitudinal responses since we use a generalized linear mixed model for the longitudinal process. A simulation study shows that our method gives good statistical performance and highlights the importance of taking into account the possible nonlinear effects of certain survival covariates. Data from patients with a first progression of glioblastoma are analysed to illustrate the method.

Fuzzy Observer-Based Command Filtered Adaptive Control of Flexible Joint Robots With Time-Varying Output Constraints

Fuzzy Observer-Based Command Filtered Adaptive Control of Flexible Joint Robots With Time-Varying Output Constraints

The single port robotic surgical "toolbox": a primer for beginners.

The aim of this study is to provide a comprehensive overview of the da Vinci Single Port robotic platform, including instruments and tools that can aid in implementing the use of this novel platform. Footage recorded during various Single port robotic urologic procedures and dry labs performed at two US institutions was used as video material. A step-by-step guide illustrating key points on OR set-up, platform, instruments, trocar configurations, intraoperative suctioning, bedside assistance were discussed and highlighted. The Single port surgeon console resembles the Xi console but includes upgraded software. The 6-mm biarticulated instruments incorporate an elbow and a wrist flexible joint. These instruments are deployed through the Access port. Access port kit includes the Access port, and a 25-mm multichannel trocar accommodating an 8-mm flexible scope, and three 6-mm robotic instruments. The 0° endoscope has two sets of articulation: a fixed one, and a distal one, allowing for three movements, selected with a hand command, the "Camera Adjust", the "Camera Control" and the "Relocation." The "Cobra mode," is an extra setting that allows the camera to wing out and move laterally relative to the working instruments. Suction is preferably performed with the Remotely Operated Suction Irrigation system. Herein we provide a detailed guide to the main technical nuances of the Single port platform and a practical overview of the instrumentation that is used during Single port robotic procedures. Knowledge of the toolbox that is used during Single port robotic surgery is key for those approaching for the first time this novel technology.

Local joint flexibility of two-planar tubular DY-joints in jacket-type substructures of offshore wind turbines: parametric study of geometrical effects and design formulation

ABSTRACT The present paper discusses the results of a parametric study conducted on the local joint flexibility (LJF) of two-planar tubular DY-joints in jacket-type substructures of offshore wind turbines. A set of finite element (FE) analyses were carried out on 81 FE models subjected to two types of axial loading to study the effects of geometrical characteristics of two-planar DY-joint on the LJF factor (f LJF). The f LJF values in two-planar DY- and uniplanar Y-joints were compared. Results showed that the multi-planarity effect on the LJF is significant and consequently the application of equations already available for uniplanar Y-joints to calculate the f LJF in two-planar DY-joints may result in highly over- or under-predicting results. To tackle this problem, FE results were used to develop a new parametric equation for the prediction of the f LJF in axially loaded two-planar DY-joints and the derived equation was checked against the UK DoE acceptance criteria.

Contrast-Enhanced Ultrasound Reveals Superior Gluteus Medius Muscle Vitality After Total Hip Arthroplasty in a Minimally Invasive Anterolateral Versus Lateral Surgical Approach

ObjectiveTotal hip arthroplasty (THA) is essential for treating severe osteoarthritis. With various surgical approaches available, there has been a shift toward techniques that offer quicker recovery and fewer complications. In this study, contrast-enhanced ultrasound was used to evaluate the impact of the minimally invasive anterolateral versus conventional lateral approach in THA on the gluteus medius muscle, focusing on muscle vitality and functional outcomes. MethodsA retrospective, cross-sectional analysis of 64 patients who underwent unilateral THA (26 lateral and 38 anterolateral) was conducted. Muscle strength was measured alongside joint flexibility and patient-reported outcome measures. Contrast-enhanced ultrasound (CEUS) examinations were conducted to visualize gluteus medius microperfusion. CEUS data were analyzed using VueBox 7.1 software and microperfusion parameters were calculated using peak enhancement (PE), wash-in perfusion index (WiPI) and rise time for muscle vitality evaluation. ResultsPatients from the anterolateral group exhibited higher relative muscle strength in the operated hip compared with the healthy side (96 ± 23% vs. 86 ± 23%, p = 0.048). CEUS revealed superior gluteus medius microperfusion in the anterolateral group (PE 36,300 ± 42,000 arbitrary units [a.u.] vs. 20,400 ± 20,200 a.u., p = 0.024; WiPI 23,600 ± 27,300 a.u. vs. 13,500 ± 13,300 a.u., p = 0.027). A positive Trendelenburg sign was evident in 2 out of 26 patients in the lateral group compared with none in the anterolateral group (χ2 = 0.082), indicating higher rates of gluteal insufficiency in the lateral approach group. ConclusionSignificantly higher muscle perfusion parameters in the anterolateral group, alongside improved muscle strength recovery, hint at a subtle but important advantage regarding post-operative recovery of the anterolateral approach. These findings support the ongoing trend toward minimally invasive surgical approaches in THA.

Seismic performance of a novel concrete-filled steel tube column to continuous reinforced concrete beam joint with multiple openings in the core region

This study developed a novel concrete-filled steel tube column to continuous reinforced concrete beam joint with multiple openings in the core region, aiming to eliminate field welding and improve construction efficiency. To evaluate the seismic performance of the joint, six full-scale specimens were tested under cyclic loading, which and a series of numerical simulations were implemented upon refined models. The effects of reinforcement ratio, steel tube thickness, and concrete strength on the failure mode and the mechanical performance degradation of the joint were quantitively evaluated. The results indicated that depending on the beam-column flexural capacity ratio, three typical failure modes, i.e., compression bending failure at the column end, shear failure at the joint core region, and flexural failure at the beam plastic hinge region, were observed. Among them, the flexural failure-dominated joint possessed the best energy dissipation capacity and a high ductility coefficient greater than 3.72. Moreover, increasing the thickness of steel tube within the core region can effectively compensate for the reduction of seismic performance caused by multiple openings, and a preliminary value of at least 2.5 times the thickness of column steel tube is recommended. This novel joint configuration exemplifies a balance between structural integrity, construction efficiency, and mechanical performance, rendering it a feasible alternative for frame structure in seismic-prone regions.

Effects of Sling-Suspension-Based Active Shoulder Joint Exercise on Shoulder Joint Subluxation, Pain, Muscle Strength, and Upper Limb Function in Patients with Subacute Stroke.

Background and Objectives: We investigated the effects of sling-suspension-based active shoulder joint exercise training on shoulder joint subluxation, pain, muscle strength, and upper extremity function in patients with subacute stroke. Materials and Methods: Twenty-eight patients with subacute stroke were randomly assigned to either the sling-suspension-based active shoulder joint exercise (SASE) group (n = 14) or the motorized upper extremity exercise (MUEE) group (n = 14). The SASE group actively performed shoulder joint flexion, extension, abduction, adduction, external and internal rotation, and horizontal abduction and adduction using a sling suspension system, whereas the MUEE group underwent an exercise program using a motorized upper extremity exercise machine. All participants underwent a 4-week intervention with 30 min of exercise once a day for 5 days a week. Additionally, both groups received general physical therapy and functional electrical stimulation for 30 min twice a day for 5 days a week. Shoulder joint subluxation was measured by radiographic examination before and after training, and pain was evaluated in the splenius, upper trapezius, and infraspinatus muscles using pressure parameters. In addition, a manual muscle tester was used to assess the muscle strength of the shoulder joint flexors, extensors, abductors, adductors, and external and internal rotators, and the Fugl-Mayer Assessment (FMA) and Manual Functional Test (MFT) were used to evaluate upper extremity function. Results: A significant group-time interaction was observed for pain, with F-values of F(1, 26) = 7.470, p < 0.011 for the splenius and F(1, 26) = 9.623, p < 0.005 for the upper trapezius. A significant time-group interaction was observed for the muscle strength of the shoulder, with F-values of F(1, 26) = 13.211, p < 0.001; F(1, 26) = 4.974, p = 0.035 and F(1, 26) = 9.674, p = 0.004 for flexors, abductors, and external rotators, respectively. A significant time-group interaction was observed in the FMA, with F-values of F(1, 26) = 13.243, p < 0.001. When comparing the interaction effects between time and group for MFT scores, a significant difference was observed, with F-values of F(1, 26) = 32.386, p < 0.001. Conclusions: This study confirmed that sling-suspension-based active shoulder joint exercises are effective in improving shoulder joint subluxation, pain, muscle strength, and upper extremity function in patients with subacute stroke.

UNDERSTANDING THE MULTI-SYSTEM MANIFESTATIONS OF HYPERMOBILITY IN YOUR PATIENTS

Hypermobility Spectrum Disorder (HSD or hEDS) is attributed to a collagen abnormality associated with excessive joint flexibility. Approximately 90% of females with hip dysplasia have hypermobility. Manifestations of hypermobility in various body systems are unique to every patient, affecting different tissues of the body with varying degrees of severity. Our purposes were to identify the manifestations of hypermobility across multiple body systems and to study the recognition of hypermobility in the medical literature of multiple specialties over multiple decades.A literature search of the major medical disciplines for key words associated with HSD was performed. These specialties included gastroenterology, gynecology, neurology, psychiatry, oral-maxillofacial surgery, cardiology, and orthopaedic surgery. A specialty-specific impact factor (IF) score was calculated as the percentage of research articles that referenced hypermobility as a comorbidity over all articles within that specialty. Statistical differences were identified using single factor ANOVA with significance determined at p&lt;0.05. We reviewed many published, specialty-specific manifestations of hypermobility, and describe them.All six non-orthopaedic specialties demonstrated a continually increasing relative IF ratio throughout the study period with a peak impact average of 0.22 (p&lt;0.05 compared with other time ranges). There was a 93.3% overall increase in IF scores from the 1992–1998 period to the most recent period examined (p&lt;0.05).Hypermobility is increasingly recognized as a significant health issue in multiple disciplines. Since dysplasia is associated with approximately 40% of all primary total hip arthroplasty cases, understanding the multi-system manifestations, and broad impact of hypermobility on patients, is relevant for every hip surgeon. We are expanding our research into other medical disciplines, including urology, ophthalmology, dermatology, clinical psychology, and others.

Research on circumferential joint shear damage characteristics of segments with positioning tenon through distributed fiber optic-based sensing technique

Positioning tenons have been used in more and more subway projects to improve segment assembly performance and to limit the occurrence of common problems between segments. The joints are the weak parts of the segment connection and are often damaged during construction. However, there are limited studies on the joints of segments with positioning tenons, and their damage characteristics have yet to be investigated. In this study, circumferential joint shear damage in segments with positioning tenons is studied and the evolution of the distributed damage is monitored by distributed fiber optics. First of all, the shear performance of segments with positioning tenons is investigated under different axial forces. Particularly, the damage distribution and development of segments subjected to the extrusion of the positioning tenons are analyzed through the strain signals and the macroscopic cracks during the shearing process. Furthermore, the changes in the strain signals of the distributed fiber before shear failure are analyzed. Based on the above study, the failure patterns of the inner surfaces and circumferential joint surfaces of segments with grooves are revealed. The interaction between the positioning tenons and the segments is analyzed, and the damage characteristics of the tenons are obtained. Finally, the main crack shear coefficient is proposed to evaluate the residual shear performance of segments when the macroscopic main crack occurs on the inner surface. This study aims to reveal the interaction between positioning tenons and segments and to promote the application of positioning tenons in subways. Further, it lays the foundation for optimizing the flexible joints of segments in subways and contributes to the safe construction of subway tunnels.

Systematic Elastostatic Stiffness Model of Over-Constrained Parallel Manipulators Without Additional Constraint Equations

The establishment of an elastostatic stiffness model for over constrained parallel manipulators (PMs), particularly those with over constrained subclosed loops, poses a challenge while ensuring numerical stability. This study addresses this issue by proposing a systematic elastostatic stiffness model based on matrix structural analysis (MSA) and independent displacement coordinates (IDCs) extraction techniques. To begin, the closed-loop PM is transformed into an open-loop PM by eliminating constraints. A subassembly element is then introduced, which considers the flexibility of both rods and joints. This approach helps circumvent the numerical instability typically encountered with traditional constraint equations. The IDCs and analytical constraint equations of nodes constrained by various joints are summarized in the appendix, utilizing multipoint constraint theory and singularity analysis, all unified within a single coordinate frame. Subsequently, the open-loop mechanism is efficiently closed by referencing the constraint equations presented in the appendix, alongside its elastostatic model. The proposed method proves to be both modeling and computationally efficient due to the comprehensive summary of the constraint equations in the Appendix, eliminating the need for additional equations. An example utilizing an over constrained subclosed loops demonstrate the application of the proposed method. In conclusion, the model proposed in this study enriches the theory of elastostatic stiffness modeling of PMs and provides an effective solution for stiffness modeling challenges they present.