Prior to the development of sophisticated computer numerical control (CNC), both face milling (FM) and face hobbing (FH), the two most popular technologies for bevel gear production, required cradle-type machines with diverse and complicated mechanisms. In the last two decades, however, the gear industry has replaced these traditional machines with six-axis CNC bevel gear cutting machines that have superior efficiency and accuracy. One such machine is a vertical six-axis machine with a vertical spindle arrangement, which offers two industrially proven advantages: compact design and maximum machine stiffness. The technical details of this machine, however, remain undisclosed, so this paper proposes a mathematical model that uses inverse kinematics to derive the vertical machine's nonlinear six-axis coordinates from those of a traditional machine. The model also reduces manufacturing errors by applying an effective flank correction method based on a sensitivity analysis of how slight variations in the individual machine setting coefficients affects tooth geometry. We prove the model's efficacy by first using the proposed equations to derive the nonlinear coordinates for pinion and gear production and then conducting several cutting experiments on the gear and its correction. Although the numerical illustration used for this verification is based only on FM bevel gears produced by an SGDH cutting system, the model is in fact applicable in the production of both FM and FH bevel gears.

It is important to maintain high kinematic reliability for robotic manipulators so that their positional errors are with acceptable limits at a required level of probability or reliability. This work accounts for random dimensions and random joint angles of robotic mechanisms and develops an efficient reliability method that can quickly and accurately predict the kinematic reliability of robotic manipulators. The kinematic reliability is defined by the probability that the actual position of an end-effector falls into a tolerance sphere, which is centered in the target position with its radius equal to the error tolerance. The motion error, which involves three dependent coordinates of the end-effector, is at first transformed into a function with only independent standard normal variables, and then the saddlepoint approximation is applied to compute the kinematic reliability. The transformation and the use of saddlepoint ensure high accuracy and high efficiency since no random simulation is needed. Examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Computational methods for kinematic synthesis of mechanisms for motion generation problems require input in the form of precision positions. Given the highly non-linear nature of the problem, solutions to these methods are overly sensitive to the input -- a small perturbation to even a single position of a given motion can change the topology and dimensions of the synthesized mechanisms drastically. Thus, the synthesis becomes a blind iterative process of maneuvering precision positions in the hope of finding good solutions. In this paper, we present a deep-learning based framework which manages the uncertain user input and provides the user with a higher level control of the design process. The framework also imputes the input with missing information required by the computational algorithms. The approach starts by learning the probability distribution of possible linkage parameters with a deep generative modeling technique, called Variational Auto Encoder (VAE). This facilitates capturing salient features of the user input and relating them with possible linkage parameters. Then, input samples resembling the inferred salient features are generated and fed to the computational methods of kinematic synthesis. The framework post-processes the solutions and presents the concepts to the user along with a handle to visualize the variants of each concept. We define this approach as Variational Synthesis of Mechanisms. This framework functions as a bridge between the current state of the art theoretical and computational kinematic methods and machine learning to enable designers to create practical mechanism design solutions.

A novel two degree-of-freedom (2-DOF) parallel mechanism with remote center-of-motion (RCM) is proposed for minimally invasive surgical applications in this paper. A surgical manipulator with expected three-rotation and one-translation (3R1T) outputs can be obtained by serially connecting a revolute pair (R) and a prismatic pair (P) to the mechanism. First, kinematics of the new mechanism is analyzed and the corresponding velocity Jacobin matrix is established. Then, singularity identification of the mechanism is performed based on screw theory. Further, main dimensions of the mechanism are designed, and a physical prototype is developed to verify the effectiveness of executing RCM. The proposed mechanism has relatively simple kinematics, and can obtain a noninterference and nonsingularity cone workspace with the top angle of 60 deg based on a compact structure.

Origami has provided various interesting applications in science and engineering. Appropriate representations and evaluation on crease patterns play an important role in developing an innovative origami structure with desired characteristics. However, this is generally a challenge encountered by scientists and engineers who introduce origami into various fields. As most practical origami structures contain repeated unit cells, graph products provide a suitable choice for the formation of crease patterns. Here, we will employ undirected and directed graph products as a tool for the representation of crease patterns and their corresponding truss frameworks of origami structures. Given that an origami crease pattern can be considered to be a set of directionless crease lines that satisfy the foldability condition, we demonstrate that the pattern can be exactly expressed by a specific graph product of independent graphs. It turns out that this integrated geometric-graph-theoretic method can be effectively implemented in the formation of different crease patterns and provide suitable numbering of nodes and elements. Furthermore, the presented method is useful for constructing the involved matrices and models of origami structures and thus enhances configuration processing for geometric, kinematic, or mechanical analysis on origami structures.

Least-square problems arise in multiple application areas. The numerical algorithm intended to compute offline the minimum (Euclidian)-norm approximation to an overdetermined system of linear equations, the core of least squares, is based on Householder reflections . It is self-understood, in the application of this algorithm, that the coefficient matrix is dimensionally homogeneous, i.e., all its entries bear the same physical units. Not all applications lead to such matrices, a case in point being parameter identification in mechanical systems involving rigid bodies, whereby both rotation and translation occur; the former being dimensionless and the latter bearing units of length. Because of this feature, dual numbers have found extensive applications in these fields, as they allow the analyst to include translations within the same relations applicable to rotations, on dualization 2 of the rotation equations, as occurring in the geometric, kinematic, or dynamic analyses of mechanical systems. After recalling the basic background on dual numbers and introducing reflection matrices defined over the dual ring , we obtain the dual version of Householder reflections applicable to the offline implementation of parameter identification. For the online parameter identification, recursive least squares are to be applied. We provide also the dual version of recursive least squares. Numerical examples are included to illustrate the underlying principles and algorithms.

The structural synthesis of Baranov trusses is still an open problem, although Baranov trusses are widely used in the design and analysis of mechanisms and robots. This paper proposes a systematic method for the structural synthesis of Baranov trusses. First, the definition review and the graph-form representations of Baranov trusses are proposed. Second, seven propositions on structural characteristics of Baranov trusses are concluded. Then, based on the set of constraint equations and a rigid subchain detection algorithm, a systematic method is presented to synthesize the complete set of Baranov trusses with a specified number of links. Finally, the synthesis results of contracted graphs (including valid and rigid contracted graphs) and topological graphs of Baranov trusses with up to 13 links are provided, and the synthesis methods and results between ours and the ones in the existing literature are compared and discussed in detail.

This paper presents the design methodology for a single-mobility, large surface-deployable mechanism using irregularly shaped triangular prismoid units. First, we demonstrate that the spherical shell, as the deployed profile of the large deployable mechanism, cannot be filled with identical regular-shaped triangular prismoids (truncated pyramid) without gaps, which makes the design challenging because a large set of nonidentical modules should be moved synchronously. Second, we discuss the design of a novel deployable mechanism that can be deployed onto irregularly shaped triangular prismoids, which will be used as the basic module to fill the spherical shell. Owing to high stiffness and ease of actuation, a planar scissor-shape deployable mechanism is applied. Third, we study the mobile assemblies of irregularly shaped modules in large surface-deployable mechanisms. We discover that hyper kinematic redundant constraints exist in a multiloop mechanism, making the design even more difficult. In order to address this issue, a methodology for reducing these redundant constraints is also discussed. Finally, a physical prototype is fabricated to demonstrate the feasibility of the proposed design methodology.

Benefiting from small incisions, reduced risk of infection, less pain, and fast recovery, minimally invasive surgery has shown tremendous advantages for patients. In these kinds of procedures, remote center-of-motion (RCM) mechanisms play an important role in performing operations through small incisions. Inspired by the Peaucellier–Lipkin straight-line cell, this paper presents the design and verification of a new type of planar two degree-of-freedom (DOF) RCM mechanism. A synthesized planar RCM mechanism is realized by a symmetric linkage actuated by two circular motion generators. The main merit of the proposed 2DOF RCM mechanism is its straightforward kinematics, which results in a simple control scheme. One of the candidate mechanisms, which is simple in structure and easy to fabricate, is intensively investigated. A prototype was built, on which preliminary experiments have been conducted, to verify the feasibility of the proposed new mechanism. The experimental results show that the fabricated 2DOF prototype has a nearly stationary remote center of motion. Therefore, the prototype has potential applicability in robot-assisted minimally invasive surgeries.

The paper initially reports about the properties of an expression of dual generalized inverse matrix currently available in the literature. It is demonstrated that such a matrix does not fulfill all the Penrose conditions. Hence, novel and computationally efficient algorithms/formulas for the computation of the Moore–Penrose dual generalized inverse (MPDGI) are herein proposed. The paper also contains a new algorithm for the singular value decomposition (SVD) of a dual matrix. The availability of these formulas allows the simultaneous solution of overdetermined systems of dual linear equations without requiring the traditional separation in primal and dual parts. This should prove useful for the solution of many kinematic problems. The algorithms/formulas herein deduced have been also tested on the kinematic synthesis of the constant transmission ratio RCCC spatial linkage.

A method is presented to optimize the shape and size of a passive, energy-storing prosthetic foot using the lower leg trajectory error (LLTE) as the design objective. The LLTE is defined as the root-mean-square error between the lower leg trajectory calculated for a given prosthetic foot's deformed shape under typical ground reaction forces (GRFs), and a target physiological lower leg trajectory obtained from published gait data for able-bodied walking. Using the LLTE as a design objective creates a quantitative connection between the mechanical design of a prosthetic foot (stiffness and geometry) and its anticipated biomechanical performance. The authors' prior work has shown that feet with optimized, low LLTE values can accurately replicate physiological kinematics and kinetics. The size and shape of a single-part compliant prosthetic foot made out of nylon 6/6 were optimized for minimum LLTE using a wide Bezier curve to describe its geometry, with constraints to produce only shapes that could fit within a physiological foot's geometric envelope. Given its single part architecture, the foot could be cost effectively manufactured with injection molding, extrusion, or three-dimensional printing. Load testing of the foot showed that its maximum deflection was within 0.3 cm (9%) of finite element analysis (FEA) predictions, ensuring the constitutive behavior was accurately characterized. Prototypes were tested on six below-knee amputees in India—the target users for this technology—to obtain qualitative feedback, which was overall positive and confirmed the foot is ready for extended field trials.

Rigid-body discretization of continuum elements was developed as a method for simplifying the kinematics of otherwise complex systems. Recent work on pseudo-rigid-body (PRB) models for compliant mechanisms has opened up the possibility of using similar concepts for synthesis and design, while incorporating various types of flexible elements within the same framework. In this paper, an idea for combining initially curved and straight beams within planar compliant mechanisms is developed to create a set of equations that can be used to analyze various designs and topologies. A PRB model with three revolute joints is derived to approximate the behavior of initially curved compliant beams, while treating straight beams as a special case (zero curvature). The optimized model parameter values are tabled for a range of arc angles. The general kinematic and static equations for a single-loop mechanism are shown, with an example to illustrate accuracy for shape and displacement . Finally, this framework is used for the design of a compliant constant force mechanism to illustrate its application, and comparisons with finite element analysis (FEA) are provided for validation.

Rehabilitation robots are increasingly being developed in order to be used by injured people to perform exercise and training. As these exercises do not need wide range movements, some parallel robots with lower mobility architecture can be an ideal solution for this purpose. This paper presents the design of a new four degree-of-freedom (DOF) parallel robot for knee rehabilitation. The required four DOFs are two translations in a vertical plane and two rotations, one of them around an axis perpendicular to the vertical plane and the other one with respect to a vector normal to the instantaneous orientation of the mobile platform. These four DOFs are reached by means of two RPRR limbs and two UPS limbs linked to an articulated mobile platform with an internal DOF. Kinematics of the new mechanism are solved and the direct Jacobian is calculated. A singularity analysis is carried out and the gained DOFs of the direct singularities are calculated. Some of the singularities can be avoided by selecting suitable values of the geometric parameters of the robot. Moreover, among the found singularities, one of them can be used in order to fold up the mechanism for its transportation. It is concluded that the proposed mechanism reaches the desired output movements in order to carry out rehabilitation maneuvers in a singularity-free portion of its workspace.

This paper presents a geometric method as a unified synthesis process of function generation for spatial linkages. The synthesis method utilizes the mapping relationship between spatial kinematic geometric model and two-plane projection system to transform the problem from spatial geometry to plane geometry. In this way, the synthesis process of mechanisms can be simplified through the corresponding transformation. Afterward, the line-guidance model is built up. Combining the kinematic inversion in two-plane projection system, this model can be used to realize the spatial linkages synthesis for function generation with three finite positions. Finally, revolute–sphere–sphere–revolute (RSSR) and revolute–sphere–sphere–prism (RSSP) mechanisms are offered to illustrate the application of this method.

In our previous work, we designed a three-degrees-of-freedom (3DOF) translational parallel mechanism based on a proposed design strategy. In this paper, the design strategy is further improved, and a novel spatial translation mechanism (STM) is found. The novel STM consists of a platform, a base, and six modules between the platform and the base. Each module is a passive planar 6R single-loop closed chain, and it is connected with two other modules. Meanwhile, three modules are connected to the base, and the other three modules are connected to the platform. All the connections among the modules, platform, and base are realized by revolute joints. There are no obvious limbs in the mechanism due to the complex connections. The mobility of the STM is analyzed, and the forward kinematics is investigated. To validate the effectiveness and feasibility of the design, one prototype is fabricated. At the end of the paper, we draw some conclusions and discuss the future works.

In this paper, a microsystem with prescribed functional capabilities is designed and simulated. In particular, the development of a straight line path generator micro electro mechanical system (MEMS) device is presented. A new procedure is suggested for avoiding branch or circuit problems in the kinematic synthesis problem. Then, Ball's point detection is used to validate the obtained pseudo-rigid body model (PRBM). A compliant MEMS device is obtained from the PRBM through the rigid-body replacement method by making use of conjugate surfaces flexure hinges (CSFHs). Finally, the functional capability of the device is investigated by means of finite element analysis (FEA) simulations and experimental testing at the macroscale.

This paper addresses the assembly strategy capable of deriving a family of overconstrained mechanisms systematically. The modular approach is proposed. It treats the topological synthesis of overconstrained mechanisms as a systematical derivation rather than a random search. The result indicates that a family of overconstrained mechanisms can be constructed by combining legitimate modules. A spatial four-bar linkage containing two revolute joints (R) and two prismatic joints (P) is selected as the source-module for the purpose of demonstration. All mechanisms discovered in this paper were modeled and animated with computer-aided design (CAD) software and their mobility were validated with input–output equations as well as computer simulations. The assembly strategy can serve as a self-contained library of overconstrained mechanisms.

Planar motion coordination of an unoriented line passing through a point or tangent to a conic is a well-known problem in kinematics. In Yaglom's algebraic geometry, oriented lines in a plane are represented with dual numbers. In the present paper, such algebraic geometry is applied in the kinematic synthesis of an inverted slider–crank for prescribed three and four finitely separated positions of the coupler. No previous application of Yaglom's algebraic geometry in the area of linkage kinematic synthesis is recorded. To describe the planar finite displacement of an oriented line about a given rotation pole, new dual operators are initially obtained. Then, the loci of moving oriented lines whose three and four homologous planar positions are tangent to a circle are deduced. The paper proposes the application of findings to the mentioned kinematic synthesis of the inverted slider–crank. Numerical examples show the reliability of the proposed approach. Finally, it is also demonstrated that, for a general planar motion, there is not any line whose five finitely separated positions share the same concurrency point. For the case of planar infinitesimal displacements, the same property was established in a paper authored by Soni et al. (1978, “Higher Order, Planar Tangent-Line Envelope Curvature Theory,” ASME J. Mech. Des., 101 (4), pp. 563–568.)

A kinematic model of the planetary roller screw mechanism (PRSM) is proposed, which accounts for the run-out errors of the screw, roller, nut, ring gear, and carrier, and the position errors of the nut and the pinhole in the carrier. The roller floating region, which contains all the possible positions of the roller inside the pinhole, is obtained by analyzing the axial clearances between mating thread surfaces and the radial clearance between the roller and carrier. The proposed model is based on the constraint that the set of roller floating region is not empty. Then, the additional rigid-body movement on the nut is derived and the path of motion transfer from the screw to the nut is obtained. According to the fundamental property of rigid-body kinematics, the axial velocity of the nut is derived and the transmission error of the PRSM is calculated. The proposed model is verified by comparing the calculated transmission error with experimental one. The results show that the transmission error of the PRSM with run-out and position errors is cyclic with a period corresponding to the rotation period of the screw and the magnitude of the transmission error can be much larger than the lead error of the screw. Besides, due to the run-out and position errors, the roller can move radially or transversally inside the pinhole of the carrier when the elements in the PRSM are regarded as rigid bodies.