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research-article  
Matthew E. Lynch, Soumalya Sarkar and Kurt Maute
J. Mech. Des   doi: 10.1115/1.4044228
Recent advances in design optimization have significant potential to improve function of mechanical components and systems. Coupled with additive manufacturing, topology optimization is one category of numerical methods used to produce algorithmically-generated optimized designs making a difference in mechanical design of hardware currently being introduced to the market. Unfortunately, many of these algorithms can require extensive manual setup and control, particularly of tuning parameters that control algorithmic function and convergence. This paper introduces a framework based on machine learning approaches to recommend tuning parameters to a user in order to avoid costly trial and error involved in manual tuning. The algorithm reads tuning parameters from a repository of prior, similar problems adjudged using a dissimilarity metric based on problem meta data, and refines them for the current problem using a Bayesian optimization approach. The approach is demonstrated for a simple topology optimization problem with the objective of achieving good topology optimization solution quality, and then with the additional objective of finding an optimal “trade” between solution quality and required computational time. The goal is to reduce the total number of “wasted” tuning runs that would be required for purely manual tuning. With more development, the framework may ultimately be useful on an enterprise level for analysis and optimization problems—topology optimization is one example but the framework is also applicable to other optimization problems such as shape and sizing and in high-fidelity physics-based analysis models—and enable these types of advanced approaches to be used more efficiently.
research-article  
Sangeun Oh, Yongsu Jung, Seongsin Kim, Ikjin Lee and Namwoo Kang
J. Mech. Des   doi: 10.1115/1.4044229
Deep learning has recently been applied to various research areas of design optimization. This study presents the need and effectiveness of adopting deep learning for generative design (or design exploration) research area. This work proposes an artificial intelligent (AI)-based deep generative design framework that is capable of generating numerous design options which are not only aesthetic but also optimized for engineering performance. The proposed framework integrates topology optimization and generative models (e.g., generative adversarial networks (GANs)) in an iterative manner to explore new design options, thus generating a large number of designs starting from limited previous design data. In addition, anomaly detection can evaluate the novelty of generated designs, thus helping designers choose among design options. The 2D wheel design problem is applied as a case study for validation of the proposed framework. The framework manifests better aesthetics, diversity, and robustness of generated designs than previous generative design methods.
TOPICS: Design, Optimization, Topology, Wheels, Robustness, Design methodology
research-article  
Dedy Suryadi and Harrison M. Kim
J. Mech. Des   doi: 10.1115/1.4044198
In generating a choice model to produce a good quality estimate of parameters related to product attributes, a high quality choice set is essential. However, the choice set data is often not available. This research proposes a methodology that utilizes online data and customer reviews to construct customer choice sets, in the absence of both actual choice set and customer socio-demographic data. The methodology consists of three main parts , i.e. clustering the products based on their attributes, clustering the customers based on their reviews, and constructing the choice sets based on a sampling probability scenario that relies on product clusters and purchased products. There are three scenarios proposed, i.e. Random as the baseline, Normalized, and Inverted. There are two utility functions proposed, i.e. a linear combination of product attributes only and a function that includes customer reviews as well. The methodology is implemented into two data sets of products, i.e. in-car DVD players and laptops. For both data sets, the Inverted scenario scores higher log-likelihood and adjusted R-squared values than both Random and Normalized. It implies that, in constructing their choice sets, customers are more likely to include groups of products that are rarely purchased by the people similar to themselves. As for the utility functions, the inclusion of customer reviews results in choice models with significantly better performance in most cases.
TOPICS: Laptop computers, Probability
research-article  
Glen Williams, Nicholas A. Meisel, Timothy W. Simpson and Christopher McComb
J. Mech. Des   doi: 10.1115/1.4044199
Machine learning can be used to automate common or time-consuming engineering tasks for which sufficient data already exists. For instance, design repositories can be used to train deep learning algorithms to assess component manufacturability; however, methods to determine the suitability of a design repository for use with machine learning do not exist. We provide an initial investigation towards identifying such a method by using “artificial” design repositories to experimentally test the extent to which altering properties of the dataset impacts the assessment precision and generalizability of neural networks trained on the data. For this experiment, we use a 3D convolutional neural network to estimate quantitative manufacturing metrics directly from voxel-based component geometries. Additive manufacturing (AM) is used as a case study because of the recent growth of AM-focused design repositories such as GrabCAD and Thingiverse that are readily accessible online. In this study, we focus only on material extrusion, the dominant consumer AM process, and investigate three AM build metrics: (1) part mass, (2) support material mass, and (3) build time. Additionally, we compare the convolutional neural network accuracy to that of a baseline multiple linear regression model. Our results suggest that training on design repositories with less standardized orientation and position resulted in more accurate trained neural networks and that orientation-dependent metrics were harder to estimate than orientation-independent metrics. Furthermore, the convolutional neural network was more accurate than the baseline linear regression model for all build metrics.
TOPICS: Design, Artificial neural networks, Additive manufacturing, Machine learning, Regression models, Trains, Manufacturing, Extruding, Algorithms
research-article  
Phillip D. Stevenson, Christopher A. Mattson and Eric C. Dahlin
J. Mech. Des   doi: 10.1115/1.4044161
All products have social impact. Some social impacts are commonly recognized by the engineering community, such as impacts to a user’s health and safety, while other social impacts are less recognized, such as impacts on families and gender roles. When engineers make design decisions, without considering social impacts, they can unknowingly cause negative social impacts. Even harming the user and/or society. Despite its challenges, measuring a program’s or policy’s impact is common practice in social sciences. These measurements are made using indicators, which are the things observed to verify that progress is being made. While there are benefits to predicting the social impact of an engineered product, it is unclear how engineers should select indicators and build predictive social impact models that are functions of engineering parameters and decisions. This paper introduces a method for selecting social impact indicators and creating predictive social impact models that can help engineers predict and improve the social impact of their products. First, an engineer identifies the product’s users, objectives, and requirements. Then, the social impact categories related to the product are determined. From each of these categories, the engineer selects social impact indicators. Finally, models are created for each indicator to predict how a product’s parameters change these indicators. The impact categories and indicators can be translated into product requirements and performance measures to be used in product development processes. This method is used to predict the social impact of the U.S. Mexico border wall.
TOPICS: Engineers, Design, Health and safety, Product development
research-article  
Kenton B. Fillingim, Richard O. Nwaeri, Felipe Borja, Katherine Fu and Christiaan J. J. Paredis
J. Mech. Des   doi: 10.1115/1.4044160
This study offers insight into the processes of expert designers at the Jet Propulsion Laboratory (JPL) and how they use heuristics in the design process. A methodology for the extraction, classification, and characterization of heuristics is presented. Ten expert participants were interviewed to identify design heuristics used during early stage space mission design at JPL. In total, 101 heuristics were obtained, classified, and characterized. The use of interviews to extract heuristics allowed for researchers to confirm those heuristics were indeed used by designers. Through the use of post-interview surveys, participants characterized heuristics based on attributes including source/origin, applicability based on concept maturity, frequency of use, reliability, and tendency to evolve. These findings are presented, and statistically significant correlations were found between the participant perceptions of frequency of use, reliability, and evolution of a heuristic. A positive correlation was found between frequency of use and reliability, while negative correlations were found between frequency of use and evolution, and reliability and evolution. Survey results and analysis aim to identify valid attributes for assessing the applicability and value of multiple heuristics for design practice in early space mission formulation.
TOPICS: Jet propulsion, Design, Teams, Reliability, Engineering design processes
research-article  
Liye Lv, Maolin Shi, Xueguan Song, Wei Sun and Jie Zhang
J. Mech. Des   doi: 10.1115/1.4044112
Infilling strategies have been proposed for decades and are widely used in engineering problems. It is still challenging to achieve an effective trade-off between global exploration and local exploitation. In this paper, a novel decision-making infilling strategy named the Go-inspired hybrid infilling (Go-HI) strategy is proposed. The Go-HI strategy combines multiple individual infilling strategies, such as the mean square error (MSE), expected improvement (EI), and probability of improvement (PoI) strategies. The Go-HI strategy consists of two major parts. In the first part, a tree-like structure consisting of several subtrees is built. In the second part, the decision value for each subtree is calculated using a cross-validation (CV)-based criterion. Key factors that significantly influence the performance of the Go-HI strategy, such as the number of component infilling strategies and the tree depth, are explored. Go-HI strategies with different component strategies and tree depth are investigated and also compared with four baseline adaptive sampling strategies through three numerical functions and one engineering case. Results show that the number of component infilling strategies exerts a larger influence on the global and local performance than the tree depth; the Go-HI strategy with two component strategies performs better than the ones with three; the Go-HI strategy always outperforms the three component infilling strategies and the other four benchmark strategies in global performance and robustness, and saves much computational cost.
TOPICS: Decision making, Errors, Probability, Robustness, Tradeoffs
research-article  
Zhen Hu, Zissimos P. Mourelatos, David Gorsich, Paramsothy Jayakumar and Monica Majcher
J. Mech. Des   doi: 10.1115/1.4044111
The Next Generation NATO Reference Mobility Model (NG-NRMM) plays a vital role in vehicle mobility prediction and mission planning. The complicated vehicle-terrain interactions and the presence of heterogeneous uncertainty sources in the modeling and simulation (M&S) result in epistemic uncertainty/errors in the vehicle mobility prediction for given terrain and soil conditions. In this paper, the uncertainty sources that cause the uncertainty in mobility prediction are first partitioned into two levels, namely uncertainty in the M&S and uncertainty in terrain and soil maps. With a focus on the epistemic uncertainty in the M&S, this paper presents a testing design optimization framework to effectively reduce the uncertainty in the M&S and thus increase the confidence in generating off-road mobility maps. A Bayesian updating approach is developed to reduce the epistemic uncertainty/errors in the M&S using mobility testing data collected under controllable terrain and soil conditions. The updated models are then employed to generate off-road mobility maps for any given terrain and soil maps. Two types of design strategies, namely testing design for model selection and testing design for uncertainty reduction, are investigated in the testing design framework to maximize the information gain subject to limited resources. Results of a numerical example demonstrate the effectiveness of the proposed mobility testing design optimization framework.
TOPICS: Optimization, Testing, Design, Roads, Uncertainty, Mechanical admittance, Soil, Vehicles, Errors, Modeling, Simulation
research-article  
Wenrui Liu, Jianwei Sun and Jinkui Chu
J. Mech. Des   doi: 10.1115/1.4044110
An open path synthesis method for a spatial revolute-revolute-spherical-spherical (RRSS) mechanism is presented in this paper. The mathematical model for the trajectory curve is established. The characteristics of an RRSS mechanism in a standard installation position are revealed: the projection points of the coupler curve on the Oxy plane rotate by the corresponding input angles around the z-axis, and the generated points lie on an ellipse. Based on this finding, a seventeen-dimensional path generation problem can be translated into two lower-dimensional matching recognition problems and one actual size and installation position calculation problem. The path generation can be achieved by three steps. First, a database of four dimensional rotation angle parameters is established. By comparing the similarities between the mechanism feature curve of the prescribed open curve and its corresponding mechanism feature ellipse, the angles of installation, the initial angle of the input link and the elliptic feature parameters of the desired RRSS mechanism can be approximately determined. Then, a thirteen-dimensional dynamic self-adapting numerical atlas database is established, which contains six basic dimensional types and seven wavelet feature parameters, and the basic dimensional types of the desired RRSS mechanism are obtained. Finally, based on the relationship between the mechanism feature ellipse of the prescribed curve and the basic dimensional types of the desired RRSS mechanism, the calculation models for the actual link lengths and installation positions of the desired RRSS mechanism were established. Three examples are presented in this paper.
TOPICS: Rotation, Trajectories (Physics), Databases, Wavelets
research-article  
Ahmed H. Bayoumy and Michael Kokkolaras
J. Mech. Des   doi: 10.1115/1.4044109
We consider the problem of selecting among different physics-based computational models of varying, and oftentimes not assessed, fidelity for evaluating the objective and constraint functions in numerical design optimization. Typically, higher-fidelity models are associated with higher computational cost. Therefore, it is desirable to employ them only when necessary. We introduce a relative adequacy framework that aims at determining whether lower-fidelity models (that are typically associated with lower computational cost) can be used in certain areas of the design space as the latter is being explored during the optimization process. The proposed approach is implemented in the mesh adaptive direct search derivative-free optimization algorithm using a trust-region management framework. We demonstrate the link between feasibility and fidelity and the key features of the proposed approach using the design example of a cantilever flexible beam subject to high accelerations.
TOPICS: Design, Optimization, Cantilevers, Optimization algorithms, Physics
research-article  
Philip Odonkor and Kemper Lewis
J. Mech. Des   doi: 10.1115/1.4044077
The flexibility afforded by distributed energy resources in terms of energy generation and storage has the potential to disrupt the way we currently access and manage electricity. But as the energy grid moves to fully embrace this technology, grid designers and operators are having to come to terms with managing its adverse effects, exhibited through electricity price volatility, caused in part by the intermittency of renewable energy. With this concern however comes interest in exploiting this price volatility using arbitrage – the buying and selling of electricity to profit from an price imbalance – for energy cost savings for consumers. To this end, this paper aims to maximize arbitrage value through the data-driven design of optimal operational strategies for distributed energy resources (DERs). Formulated as an arbitrage maximization problem using design optimization principles, and solved using reinforcement learning, the proposed approach is applied towards shared DERs within multi-building residential clusters. We demonstrate its feasibility across three unique building clusters, observing notable energy cost reductions over baseline values. This highlights a capability for generalized learning across multiple building clusters and the ability to design efficient arbitrage policies for energy cost minimization. The scalability of this approach is studied using two test cases, with results demonstrating an ability to scale with relatively minimal additional computational cost, and an ability to leverage system flexibility towards cost savings.
TOPICS: Design, Distributed power generation, Volatility, Renewable energy, Storage, Energy generation, Optimization
research-article  
Wei Chen and Mark Fuge
J. Mech. Des   doi: 10.1115/1.4044076
Real-world designs usually consist of parts with inter-part dependencies, i.e., the geometry of one part is dependent on one or multiple other parts. We can represent such dependency in a part dependency graph. This paper presents a method for synthesizing these types of hierarchical designs using generative models learned from examples. It decomposes the problem of synthesizing the whole design into synthesizing each part separately but keeping the inter-part dependencies satisfied. Specifically, this method constructs multiple generative models, the interaction of which is based on the part dependency graph. We then use the trained generative models to synthesize or explore each part design separately via a low-dimensional latent representation, conditioned on the corresponding parent part(s). We verify our model on multiple design examples with different inter-part dependencies. We evaluate our model by analyzing the constraint satisfaction performance, the synthesis quality, the latent space quality, and the effects of part dependency depth and branching factor. This paper's techniques for capturing dependencies among parts lay the foundation for learned generative models to extend to more realistic engineering systems where such relationships are widespread.
TOPICS: Design, Engineering systems and industry applications, Geometry
research-article  
Jaime Juan, M. Núria Salán, Arlindo Silva and Jose A. Tornero
J. Mech. Des   doi: 10.1115/1.4044107
The world is changing and demanding stronger, lighter and more versatile materials. Taking advantage of the full potential of these materials also requires versatile manufacturing processes. The in situ forming of a liquid infused preform (ISFLIP) is a new manufacturing process for fiber reinforced polymer (FRP) parts with shell shapes. ISFLIP is a hybrid process between vacuum infusion (VI) and diaphragm forming. This paper focuses on the mechanical design and experimental validation of a functional prototype of ISFLIP. The novelty of the design lies especially in a double diaphragm system that is fundamental to carrying out the forming just after the infusion stage. The double diaphragm system and other two major subsystems, a vacuum table and an infrared heating grid, were devised to benefit from the operational advantages of ISFLIP. The whole prototype, once constructed, was tested by forming some demonstration components. The result of one of these components, a ‘C’ cross-section FRP profile with two sharp joggles, is finally obtained, proving the feasibility of the prototype.
TOPICS: Engineering prototypes, Design, Preforms, Diaphragms (Mechanical devices), Diaphragms (Structural), Vacuum, Manufacturing, Fiber reinforced plastics, Fibers, Shapes, Shells, Heating, Design engineering, Polymers
research-article  
Jize Zhang and Alexandros Taflanidis
J. Mech. Des   doi: 10.1115/1.4044005
This paper presents a surrogate model based computationally efficient optimization scheme for design problems with multiple, probabilistic objectives estimated through stochastic simulation. It examines the extension of the previously developed MODU-AIM (Multi-Objective Design under Uncertainty with Augmented Input Metamodels) algorithm, which performs well for bi-objective problem, but encounters scalability difficulties for applications with more than two objectives. Computational efficiency is achieved by using a single surrogate model, adaptively refined within an iterative optimization setting, to simultaneously support the uncertainty quantification and the design optimization, and the MODU-AIM extension is established by replacing the originally used epsilon-constraint optimizer with multi-objective evolutionary algorithms (MOEA). This requires various modifications to accommodate MOEA's unique traits. For uncertainty quantification, a clustering-based importance sampling density selection is introduced to mitigate MOEA's lack of direct control on Pareto solution density. To address the potentially large solution set of MOEAs, both the termination criterion of the iterative optimization scheme and the design of experiment (DoE) strategy for refinement of the surrogate model are modified, leveraging efficient performance comparison indicators. The importance of each objective in the different parts of the Pareto front is further integrated in the DoE to improve the adaptive selection of experiments.
TOPICS: Density, Simulation, Algorithms, Design, Optimization, Evolutionary algorithms, Experimental design, Pareto optimization, Mobile offshore drilling units, Uncertainty, Design under uncertainty, Uncertainty quantification
research-article  
John K. Ostrander, Conrad S. Tucker, Timothy W. Simpson and Nicholas A. Meisel
J. Mech. Des   doi: 10.1115/1.4044006
Limited academic course offerings and high barriers to incorporate industrial Additive Manufacturing (AM) systems into education has led to an underserved demand for a highly skilled AM workforce. In this research, virtual reality (VR) is proposed as a medium to help teach introductory concepts of AM in an interactive, scalable manner. Before implementing VR as a standard tool to teach introductory concepts of AM, we must evaluate the effectiveness of this medium for the subject. We test the hypothesis that VR can be used to teach students introductory concepts of AM in a way that is as effective as teaching the same concepts in a real-world physical setting. The research also explores differences in learning between participants who engage in a hands-on interactive lesson and participants who engage in a hands-off passive lesson. The study assesses participants' AM knowledge through pre-/post-AM lesson evaluation. AM conceptual knowledge gained and changes in self-efficacy are evaluated to make an argument for the effectiveness of VR as an AM learning tool. Our findings in this research indicate that both interactive and passive VR may indeed be used to effectively teach introductory concepts of AM; we also found advantages to using interactive VR for improving AM self-efficacy.
TOPICS: Virtual reality, Additive manufacturing, Teaching, Students, Education
research-article  
Rajneesh Kumar Rai and Sunil Punjabi
J. Mech. Des   doi: 10.1115/1.4043934
Isomorphism (structural similarity) of kinematic chains (KCs) of mechanisms is an important issue in the structural synthesis, which must be identified to avoid the duplicate structures. Duplication causes incorrect family size i.e. distinct KCs with a given number of links (n) and degree of freedom (dof). Besides simple joints kinematic chains (SJKCs), multiple joints kinematic chains (MJKCs) are also widely used because of their compact size and the methods dealing with such KCs are few. The proposed method deals with two different structural invariants i.e. primary structural invariants (provide only the necessary condition of isomorphism) such as link connectivity number (LCN) of all the links, link connectivity number of chain (CCN), joint connectivity number (JCN) of all the joints and joint connectivity number of chain (JCNC) and secondary structural invariants (provide the sufficient condition of isomorphism) such as power transmission (P) and transmission efficiency (Te). Primary structural invariants are calculated using a new link-link connectivity matrix (LLCM), whereas secondary structural invariants are calculated using the concept of entropy of information theory. The method has been successfully tested for 10 and 11 links MJKCs (illustrative examples taken in the paper) and for the families of 18 MJKCs with 8 links, 2 MJs, 1-dof and 3 independent loops, 22 MJKCs with 8 links, 1 MJ, 1-dof and 3 independent loops and 83 MJKCs with 9 links, 1 MJ, 2-dof and 3 independent loops.
TOPICS: Kinematic chains, Chain, Entropy, Degrees of freedom, Cloud condensation nuclei
research-article  
Lewei Tang, Pengshuai Shi, Li Wu, Xiaoyu Wu and Xiaoqiang Tang
J. Mech. Des   doi: 10.1115/1.4043937
This paper presents a singularity study on a special class of spatial Cable-Suspended Parallel Mechanisms (CSPMs) with merely three translational degrees of freedom using redundant actuators. This paper focuses on the CSPMs which have the capability to perform the purely translational movement with pairwise cables as parallelograms. There are two types of singularity to be discussed, which result from dynamic equations of CSPMs and the parallelogram constraint of pairwise cables. To ensure three-translational dofs without rotation of the end-effector, the matrix formed by normals of the planes based on each pairwise cables should maintain in full rank. In the case study, four typical designs of CSPMs with a planar end-effector and a spatial end-effector are discussed to clarify and conclude the singularity features of CSPMs with actuation redundancy. The results show that for some architectures there exist both types of singularity for redundantly actuated CSPMs with pairwise cables but for some other architectures the redundant actuation exerts no effect on the singularity issue.
TOPICS: Cables, Redundancy (Engineering), Parallel mechanisms, End effectors, Architecture, Rotation, Degrees of freedom, Actuators, Equations of motion
research-article  
Dario Richiedei and Alberto Trevisani
J. Mech. Des   doi: 10.1115/1.4043936
The ever-growing interest towards energy efficiency imposes the optimization of mechanism design under an energetic point of view. Even if the benefit of using spring balancing systems to reduce energy consumption is intuitive, the relation between spring design and electrical energy consumption has never been systematically addressed in the literature, which is mainly focused on static compensation of gravity forces. This paper tackles this novel and important issue and proposes an analytical method for model-based design of springs minimizing the energy required in rest-to-rest motion. The method relies on the model of energy dissipation that accounts for the characteristics of the mechanical, electrical and power electronic components of a servo-actuated mechanism. The theory is developed with reference to a single rotating beam. The proposed solution ensures significant energy saving compared with the traditional static balancing design of springs and is particularly suitable for repetitive (cyclic) motion tasks.
TOPICS: Servomechanisms, Optimization, Energy consumption, Springs, Design, Electronic components, Energy dissipation, Energy efficiency, Rotating beams, Gravity (Force)
research-article  
ZHIYANG YU, Kristina Shea and Tino Stanković
J. Mech. Des   doi: 10.1115/1.4043931
The main limitations of currently available artificial spinal discs are geometric unfit and unnatural motion. Multi-material Additive Manufacturing (AM) offers a potential solution for the fabrication of personalized free-form implants with better fit and variable material distribution to achieve a set of target physiological stiffnesses. The structure of the artificial spinal disc proposed in this paper is inspired from a natural disc and includes both a matrix and a crisscross fiber-like structure, where the design variables are their material properties. After carrying out design variable reduction using linking strategies, a finite-element based optimization is then conducted to calculate the optimized material distribution to achieve physiological stiffness under five loading cases. The results show a good match in stiffness of the multi-material disc compared to the natural disc and that the multi-material artificial disc outperforms a current known solution, the ball-and-socket disc [1][2][3]. Moreover, the potential of achieving an improved match in stiffness with a larger range of available 3D printable materials is demonstrated. Although the direct surgical implantation of the design is hindered currently by the biocompatibility of the 3D printed materials, a potential improvement of the design proposed is shown.
TOPICS: Design, Disks, Stiffness, Physiology, Additive manufacturing, Computational methods, Fibers, Manufacturing, Materials properties, Biocompatibility, Finite element analysis, Optimization, Surgery
research-article  
Chen Zhao, Ziming Chen, Y.W. Li and Zhen Huang
J. Mech. Des   doi: 10.1115/1.4043938
In this paper, a novel 3-UPU (P and U stand for prismatic and universal joints, respectively) parallel mechanism (PM) and its variant PM are proposed. Either of them has two rotational and one translational (2R1T) degrees of freedom (DOFs) and has no parasitic motion. The mobility analysis shows that three constraint forces provided by three limbs of the mechanism are located on the same plane and the mobile platform is capable of translating perpendicular to this plane and rotating around any axis on this plane. The motion characteristics of the mechanism are disclosed that the output motion of the mobile platform is only pure rotation or pure translation. And the rotation axis is constant during a rotation which means no parasitic motion. The reasons are analyzed and explained by the overall Jacobian matrix, statistical method and geometric method. Besides, it only needs to translate or rotate once to move from the initial configuration to the final configuration which facilitates the speed control. Then, the relationship between mechanism parameters and singularity is analyzed. At last, we proposed the speed control method for the PMs in this paper. And the prototype of the mechanism is manufactured. Then, the experiment is conduct that verifies the motion characteristics, the speed control method by overall Jacobian matrix and singularity.
TOPICS: Parallel mechanisms, Rotation, Jacobian matrices, Mechanical admittance, Universal joints, Engineering prototypes, Degrees of freedom

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