Kinematic synthesis to meet an approximate motion specification naturally forms a constrained optimization problem. In this work, we conduct global design searches by direct computation of all critical points through stationarity conditions. The idea is not new, but performed at scale is only possible through modern polynomial homotopy continuation solvers. Such a complete computation finds all minima, including the global, serving as a powerful design exploration technique. We form equality constrained objective functions that pertain to the synthesis of spherical four-bar linkages for approximate function generation. For each problem considered, Lagrangian stationarity conditions set up a square system of polynomials. We consider the most general case where all mechanism dimensions may vary, and a more specific case that enables the placement of ground pivots. The former optimization problem is shown to permit an estimated maximum of 268 sets of critical points, and the latter permits 61 sets. Critical points are classified as saddles or minima through a post-process eigenanalysis of the projected Hessian. Approximate motion is specified as discretized points from a desired input-output angle function. The coefficients of the stationarity polynomials can be expressed as summations of symmetric matrices indexed by the discretization points. We take the sums themselves to parameterize these polynomials rather than constituent terms (the discrete data). In this way, the algebraic structure of the synthesis equations remains invariant to the number of discretization points chosen. The results of our computational work were used to design a mechanism that coordinates the unfolding of wings for a deployable aircraft.
This content will become publicly available on October 1, 2024
- Award ID(s):
- 2041789
- NSF-PAR ID:
- 10483401
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Mechanism and Machine Theory
- Volume:
- 188
- Issue:
- C
- ISSN:
- 0094-114X
- Page Range / eLocation ID:
- 105310
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
null (Ed.)Abstract Cognate linkages are mechanisms that share the same motion, a property that can be useful in mechanical design. This article treats planar curve cognates, that is, planar mechanisms with rotational joints whose coupler points draw the same curve, as well as coupler cognates and timed curve cognates. The purpose of this article is to develop a straightforward method based solely on kinematic equations to construct cognates. The approach computes cognates that arise from permuting link rotations and is shown to reproduce all of the known results for cognates of four-bar and six-bar linkages. This approach is then used to construct a cognate of an eight-bar and a ten-bar linkage.more » « less
-
more » « less
Abstract This paper focuses on the representation and synthesis of coupler curves of planar mechanisms using a deep neural network. While the path synthesis of planar mechanisms is not a new problem, the effective representation of coupler curves in the context of neural networks has not been fully explored. This study compares four commonly used features or representations of four-bar coupler curves: Fourier descriptors, wavelets, point coordinates, and images. The results demonstrate that these diverse representations can be unified using a generative AI framework called variational autoencoder (VAE). This study shows that a VAE can provide a standalone representation of a coupler curve, regardless of the input representation, and that the compact latent dimensions of the VAE can be used to describe coupler curves of four-bar linkages. Additionally, a new approach that utilizes a VAE in conjunction with a fully connected neural network to generate dimensional parameters of four-bar linkage mechanisms is proposed. This research presents a novel opportunity for the automated conceptual design of mechanisms for robots and machines.
-
Abstract Cognate linkages provide the useful property in mechanism design of having the same motion. This paper describes an approach for determining all coupler curve cognates for planar linkages with rotational joints. Although a prior compilation of six-bar cognates due to Dijksman purported to be a complete list, that analysis assumed, without proof, that cognates only arise by permuting link rotations. Our approach eliminates that assumption using arguments concerning the singular foci of the coupler curve to constrain a cognate search and then completing the analysis by solving a precision point problem. This analysis confirms that Dijksman’s list for six-bars is comprehensive. As we further demonstrate on an eight-bar and a ten-bar example, the method greatly constrains the set of permutations of link rotations that can possibly lead to cognates, thereby facilitating the discovery of all cognates that arise in that manner. However, for these higher order linkages, the further step of using a precision point test to eliminate the possibility of any other cognates is still beyond our computational capabilities.more » « less
-
Abstract This paper presents a design methodology for mechanisms consisting of a single continuous structure, continuum mechanisms, that blends the kinematic synthesis of rigid-body mechanisms with topology optimization for compliant mechanisms. Rather than start with a generic structure that is shaped to achieve a required force deflection task for a compliant mechanism, our approach shapes the initial structure based on kinematic synthesis of a rigid body mechanism for the required movement, then the structure is shaped using Finite Element Analysis to achieve the required force deflection relationship. The result of this approach is a continuum mechanism with the same workpiece movement as the rigid link mechanism when actuated. An example illustrates the design process to obtain an eight-bar linkage that guides its workpiece in straight-line rectilinear movement. We show that the resulting continuum mechanism provides the desired rectilinear movement. A 210 mm physical model machined from Nylon-6 is shown to achieve 21.5mm rectilinear movement with no perceived deviation from a straight-line.more » « less
