Khalid Al-Widyan

A Model-Based Formulation of Robust Design

Laid down in this paper are the foundations on which the design of engineering systems, in the presence of an uncontrollable changing environment, can be based. The changes in environment conditions are accounted for by means of robustness. To this end, a theoretical framework as well as a general methodology for model-based robust design are proposed. Within this framework, all quantities involved in a design task are classified into three sets: the design variables (DV), grouped in vectorx, which are to be assigned values as an outcome of the design task; the design-environment parameters (DEP), grouped in vector p, over which the designer has no control; and the performance functions (PF), grouped in vectorf, representing the functional relations among performance, DV, and DEP. A distinction is made between global robust design and local robust design, this paper focusing on the latter. The robust design problem is formulated as the minimization of a norm of the covariance matrix of the variations in PF upon variations in the DEP, aka noise in the literature on robust design. Moreover, one pertinent concept is introduced: design isotropy. We show that isotropic designs lead to robustness, even in the absence of knowledge of the statistical properties of the variations of the DEP. To demonstrate our approach, a few examples are included.@DOI: 10.1115/1.1829728# Robust design has been around for about half of a century by now. In fact, Genichi Taguchi, the recognized father of the concept, developed his approach in the postwar years, in which design engineering was based on experience and extremely costly experiments performed on a black box. The engineer could apply known inputs to the black box under various laboratory-emulated operation conditions and measure the performance of the good under design @1#. The outcome of these measurements would then be recorded in an orthogonal array @2#. These are tableaux relating recorded variations in the environment parameters with the corresponding variations in performance indicators. Indeed, Taguchi’s design paradigm is based on two fundamental concepts: ~a! the signal-to-noise ~S/N! ratio, that measures the sensitivity of a design to changes in environment conditions—the larger the S/N ratio, the more robust the performance; and ~b! the loss function, that measures the loss of society by virtue of a flawed design. As Taguchi put forth, the roots of poor quality in goods or services are to be found in the sensitivity of these to variations in operation conditions, which lie beyond the control of those in charge of the whole production process, from design to marketing and distribution. The key to high-quality goods and services is, thus, the minimization of the said sensitivity @3#. In fact, the performance sensitivity to environment changes cannot be eliminated. The aim of robust design is, thus, not to eliminate the sensitivity at stake, but rather to minimize it. Current engineering design is largely model-based, while the need for goods and services that are as insensitive as possible to environmental changes is growing, mostly because of the need to design reliable, accurate multidegree-of-freedom systems. Consequently, over the past decades, robust design has attracted the attention of researchers and practitioners from the design community. As a result, several approaches have been proposed in the literature on robust design and a wide range of applications has been reported. To be sure, the aspects of the validity of Taguchi’s method were brought into question by Box @4#, the concepts of the loss function and the signal-to-nose ratio in connection with parameter design having been critically ascertained in this reference. Other researchers, in turn, have engaged in extending Taguch’s method to allow for constrained engineering design problems, which arguably cannot be handled with the current Taguchi methods @5#. With the aid of nonlinear programming, robust design has also been formulated as worst-case scenarios, in which both the robustness of design objectives and constraints are considered @6#. Another research mainstream is oriented toward the response surface methodology and the compromise decision support problem

A Numerically Robust Algorithm to Solve the Five-Pose Burmester Problem

Introduced in this paper is a robust algorithm to solve the five-pose planar Burmester problem. The proposed algorithm functions even in the presence of special conditions of the prescribed poses that lead to algorithmic singularities otherwise. In order to show the applicability and to validate the robustness of the proposed algorithm, some examples are included.Copyright

The Robust Synthesis of Planar Four-Bar Linkages for Motion Generation

The robust synthesis of motion generators in the realm of the Burmester problem is discussed in this paper. The synthesis procedure regards the original problem as a pick-and-place operation, where the intermediate poses are defined so as to produce a linkage that is least sensitive to variations in these poses.Copyright

The Robust Design of Schönflies-Motion Generators

The subject of this paper is the robust design of a four-degree-of-freedom manipulator of the SCARA type, i.e., a Schonflies-motion generator. The optimally robust design is secured by means of mechanical isotropy, which thus leads to robustness in the main aspects of the performance, namely, kinetostatic, elastostatic, and elastodynamic. It is shown that a mechanically isotropic design is possible for a parallel manipulator.

The Kinematic Synthesis of a Robust RCCC Mechanism for Pick-and-Place Operations

Proposed in this paper is a methodology to synthesize a RCCC four-bar linkage intended for pick-and-place operations. The synthesis problem is set in the context of rigid-body guidance, which can be solved exactly for up to four prescribed poses of the coupler link. As a consequence, for a pick-and-place operation, the selection of the unspecified two intermediate poses is thus left up to the mechanism designer’s judgment. In this paper, we propose a method to determine the two intermediate poses resorting to the concept of robustness. In fact, robustness is needed in this context to overcome the presence of uncertainty due to the selection of the two unspecified poses. To this end, a theoretical framework for model-based robust design is invoked and a general methodology for robust kinematic synthesis is laid down. A numerical example is included to validate the concepts and illustrate the application of the methodology proposed here.Copyright

The Robust Linkage Synthesis for Planar Rigid-Body Guidance

The kinematic synthesis of planar motion generators in the presence of an incomplete set of finitely separated poses is the subject of this paper. Given that the planar rigid-body guidance problem in the realm of four-bar linkage synthesis can be solved exactly for up to five prescribed poses of the coupler link, any number of poses smaller than five is considered incomplete in this paper. The poses completing the set are determined so as to produce a robust linkage against variations in the unspecified poses. To this end, a theoretical framework for model-based robust design is invoked and a general methodology for robust kinematic synthesis is laid down. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the unprescribed poses, which many a time are left up to the mechanism designer’s judgment. To validate the concepts and illustrate the application of the methodology proposed here, an example is included.Copyright

The Synthesis of Robust Spherical Motion Generators

The synthesis of spherical motion generators in the presence of an incomplete set of finitely-separated attitudes is discussed in this paper. Given that five attitudes of the coupler link define a discrete set of dyads, any number of attitudes smaller than five is considered incomplete in this paper. The attitudes completing the set are determined so as to produce a robust linkage against variations in these attitudes. To this end, a theoretical framework as well as a general methodology for robust synthesis are laid down. Robustness is needed in this context to overcome the presence of uncertainty due to the selection of the intermediate attitudes, which many a time are left up to the mechanism designer’s judgment. To validate the concepts and illustrate the application of the methodology proposed here, we include a numerical example.Copyright

A Robust Solution to the Spherical Rigid-Body Guidance Problem

In this paper we introduce a robust algorithm to solve the five-attitude spherical Burmester problem associated with exact linkage synthesis. The proposed algorithm solves for the unit vectors determining the four joint centers of the linkage while taking into account all redundant equations available, which enhances the robustness of the algorithm. In order to show the applicability of the proposed algorithm and to validate its robustness, two examples are included.Copyright

A Model-Based Framework for Robust Design

In this paper we develop an as-yet-missing theoretical framework as well as a general methodology for model-based robust design. At the outset, a distinction is made between three sets: the set of design variables, grouped in the n-dimensional vector x, which are to be assigned values as an outcome of the design job; the set of design-environment parameters (DEP), grouped in the v-dimensional vector p, over which the designer has no control; and the set of performance functions, arrayed in the m-dimensional vector f, representing the functional relations among performance, design variables and DEP. Resorting to the mathematical model available for the object under design, an m × v design performance matrix F, mapping the space of relative variations of p into that of relative variations of f, is derived. Moreover, two pertinent concepts are introduced: the design sensitivity matrix, which plays a major role in the transmission of the variations of p into variations off, and its associated bandwidth, defined as the logarithm of the square root of the ratio between the maximum to the minimum singular values of the design performance matrix, measured in decades. A result stating the relation between the bandwidth of a matrix and its inverse is shown. Consequently, the aforementioned bandwidth represents an index for evaluating the robustness of a design. To demonstrate our approach, case studies are included