Synthesizing circuit-, branch-, or order-defects-free planar four-bar mechanism for the motion generation problem has proven to be a difficult problem. These defects render synthesized mechanisms useless to machine designers. Such defects arise from the artificial constraints of formulating the problem as a discrete precision position problem and limitations of the methods, which ignore the continuity information in the input. In this paper, we bring together diverse fields of pattern recognition, machine learning, artificial neural network, and computational kinematics to present a novel approach that solves this problem both efficiently and effectively. At the heart of this approach lies an objective function, which compares the motion as a whole thereby capturing designer's intent. In contrast to widely used structural error or loop-closure equation-based error functions, which convolute the optimization by considering shape, size, position, and orientation of the given task simultaneously, this objective function computes motion difference in a form, which is invariant to similarity transformations. We employ auto-encoder neural networks to create a compact and clustered database of invariant motions of known defect-free linkages, which serve as a good initial choice for further optimization. In spite of highly nonlinear parameters space, our approach discovers a wide pool of defect-free solutions very quickly. We show that by employing proven machine learning techniques, this work could have far-reaching consequences to creating a multitude of useful and creative conceptual design solutions for mechanism synthesis problems, which go beyond planar four-bar linkages.

In this paper, we have presented a unified framework for generating planar four-bar motions for a combination of poses and practical geometric constraints and its implementation in MotionGen app for Apple's iOS and Google's Android platforms. The framework is based on a unified type- and dimensional-synthesis algorithm for planar four-bar linkages for the motion-generation problem. Simplicity, high-utility, and wide-spread adoption of planar four-bar linkages have made them one of the most studied topics in kinematics leading to development of algorithms and theories that deal with path, function, and motion generation problems. Yet to date, there have been no attempts to develop efficient computational algorithms amenable to real-time computation of both type and dimensions of planar four-bar mechanisms for a given motion. MotionGen solves this problem in an intuitive fashion while providing high-level, rich options to enforce practical constraints. It is done effectively by extracting the geometric constraints of a given motion to provide the best dyad types as well as dimensions of a total of up to six four-bar linkages. The unified framework also admits a plurality of practical geometric constraints, such as imposition of fixed and moving pivot and line locations along with mixed exact and approximate synthesis scenarios.

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

This paper brings together computer vision, mechanism synthesis, and machine learning to create an image-based variational path synthesis approach for linkage mechanisms. An image-based approach is particularly amenable to mechanism synthesis when the input from mechanism designers is deliberately imprecise or inherently uncertain due to the nature of the problem. In addition, it also lends itself naturally to the creation of a unified approach to mechanism synthesis for different types of mechanisms, since for example, images are formed from a collection of pixels, which themselves could be generated from a four-bar or six-bar. Path synthesis problems have generally been solved for a set of precision points on the intended path such that the designed mechanism passes through those points. This approach usually leads to a small set of over-fitted solutions to particular precision points. However, most kinematic synthesis problems are concept generation problems, where a designer cares more about generating a large number of plausible solutions, which could reach given precision points only approximately. This paper models the input curve as a probability distribution of image pixels and employs a probabilistic generative model to capture the inherent uncertainty in the input. In addition, it gives feedback on the input quality and provides corrections for a more conducive input. The image representation allows for capturing local spatial correlations, which plays an important role in finding a variety of solutions with similar semantics as the input curve. This approach is also conducive to implementation for pressure-sensitive touch-based design interfaces, where the input is not a zero-thickness curve, but the sweep of a small patch on the finger.

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for nonlinear least squares fitting of these algebraic manifolds. This paper presents a new formulation for the manifold fitting problem using algebraic distance and shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.