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.

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.

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.

The classic Burmester problem is concerned with computing dimensions of planar four-bar linkages consisting of all revolute joints for five-pose problems. In the context of motion generation, each pose can be seen as a constraint that the coupler of a planar four-bar mechanism needs to interpolate or approximate through. We define extended Burmester problem as the one where all types of planar four-bars consisting of dyads of type RR, PR, RP, or PP (R: revolute, P: prismatic) and their dimensions need to be computed for n-geometric-constraints, where a geometric constraint can be an algebraically expressed constraint on the pose, or location of the fixed or moving pivots or something equivalent. In addition, we include both linear and non-linear and exact and approximate constraints. This extension also includes the problems where there is no solution to the classic Burmester problem, but designers would still like to design a four-bar that may come closest to capturing their intent. Such problems are representative of the problems that machine designers grapple with while designing linkage systems for a variety of constraints, which are not merely a set of poses. Recently, we have derived a unified form of geometric constraints of all types of dyads (RR, RP, PR, and PP) in the framework of kinematic mapping and planar quaternions, which map to generalized manifolds (G-manifolds) in the image space of planar displacements. The given poses map to points in the image space. Thus, the problem of synthesis is reduced to minimizing the algebraic error of fitting between the image points and the G-manifolds. We have also created a simple, two-step algorithm using Singular Value Decomposition (SVD) for the least-squares fitting of the manifolds, which yields a candidate space of solution. By imposing two fundamental quadratic constraints on the candidate solutions, we are able to simultaneously determine both the type and dimensions of the planar four-bar linkages. In this paper, we present 1) a unified approach for solving the extended Burmester problem by showing that all linear- and non-linear constraints can be handled in a unified way without resorting to special cases, 2) in the event of no or unsatisfactory solutions to the synthesis problem certain constraints can be relaxed, and 3) such constraints can be approximately satisfied by minimizing the algebraic fitting error using Lagrange Multiplier method. In doing so, we generalize our earlier formulation and present a new algorithm, which solves new problems including optimal approximate synthesis of Burmester problem with no exact solutions.

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. In addition, we also present an alternate End-to-End deep neural network architecture for Variational Synthesis of linkages. This End-to-End architecture is a Conditional-VAE (C-VAE), which approximates the conditional distribution of linkage parameters with respect to coupler trajectory distribution. The outcome is a probability distribution of kinematic linkages for an unknown coupler path or motion. 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.