Algebraic coupler curve of spherical four-bar linkages and its applications

Highlights

• An algebraic coupler curve was derived for the first time.

• The coupler curve is found as a trivariate quartic in Cartesian coordinates and a bivariate 16th degree polynomial in converted coordinates.

• New formulations were achieved by virtue of parameterized coordinates.

• Applications of the algebraic coupler curve in curvature analysis and path synthesis are elaborated.

Abstract

The derivation of coupler curve equation of spherical four-bar linkages is studied in this paper. By utilizing parameterized coordinates, an algebraic coupler curve equation was formulated for the spherical four-bar linkages, which is in the form of trivariate quartic. The algebraic coupler curve allows us to study global properties and its instantaneous properties, and also facilitates path synthesis. In the paper, the derivation of algebraic coupler curve is presented, along with different expressions of the curve. Applications of the explicit algebraic coupler curve are demonstrated with examples.

Introduction

Spherical four-bar linkages are a type of basic mechanisms to generate spherical motion, for which all links’ movements are confined in spherical surfaces [1], [2], [3], [4]. While there are increasing interests on more advanced spherical parallel manipulators for spherical motions [5], study on the spherical four-bar linkages remains active on topics including rigid-body guidance, or motion synthesis [6], [7], [8], function synthesis [9] and input-output analysis  [10], [11], [12] and curvature analysis [13].

The spherical four-bar linkage, being an one degree-of-freedom mechanism, is considered as the counterpart of the planar four-bar linkage. Thus research problems such as input-output analysis, motion synthesis, and others, can be studied uniformly for the two types of linkages [14]. An exception is mobility analysis, as a coupler curve equation similar to the planar linkages is not available in literature for spherical four-bar linkages. For planar four-bar linkages, it is well known that the algebraic coupler curve of a four-bar linkage is in a form of sextic bivariate polynomial [15], [16]. The sextic allows to determine all nine linkage parameters if the algebraic coupler curve is known [17], [18],

$$f(x,y)=\sum _{i,j=0}^6{K}_{ij}{x}^i{y}^j=0,i+j\le 6$$

where $K_{ij}$ is the coefficient of term $x^i{y}^j$. It is also known that the four-bar linkage coupler curve is a special form of sextic bivariate polynomial, which only has 15 coefficients.

However, until now, there is not an algebraic coupler curve that has been generally derived for the spherical four-bar linkages yet. An early work on the coupler curve of spherical four-bar linkage can be dated back to 1940s reported by Dobrovolskii [19]. In his work, Dobrovolskii developed a geometric method to derive the coupler curve. Following his method, a formulation of the coupler curve was presented by Wittenburg for a special case, where the coupler point is expressed as its longitude and latitude with respect to the base link [4]. Chiang presented a formulation of the coupler curve with spherical trigonometry and derived an expression in terms of specially defined coordinates called spherical rectangle coordinates [3]. Other works on the generation of coupler curves for spherical four-bar linkages adopted mainly parametric approaches, in which the input angle has to be specified and then the corresponding coupler point is calculated [20], [21]. An image curve to represent the coupler motion of a doubly folding spherical four bar linkage was reported in [22]. As the curve was generated on a hyper-sphere of the imaginary 4D Euclidean space, the curve is not displayable. While these approaches work to certain extents, they are neither efficient in curve generation, nor effective in providing an overview of the global properties of the coupler motion with respect to any varying linkage parameter.

The challenge in deriving algebraically an algebraic coupler curve lies in the spatial rigid-body rotation, which has to use three parameters of rotation, or four parameters in the case of quaternion. This brings up the problem of eliminating them to finally obtain the algebraic coupler curves. An interesting question is thus whether an algebraic equation similar to Eq.  (1) exists for spherical four-bar linkages, and if so, how to find the equation.

In this paper, we aim to develop a new method to yield an algebraic coupler curve. This is achieved by virtue of parameterized coordinates, which eliminates the need of using multiple parameters of rigid-body rotation. By this way, an algebraic coupler curve is obtained, which is the intersection of a trivariate quartic with a sphere. Upon the algebraic equation, other forms of the coupler curve, including its projective curve, curves in spherical coordinates, and curve with tangent half-angle substitutions of the latter, are derived. These equations of the spherical coupler curve allow us to reveal both its global properties and also its instantaneous properties given by the curvature and torsion functions. Moreover, the formulation with parameterized coordinates facilitates kinematic synthesis, for example, path synthesis with discrete points, five-position synthesis with prescribed timing, among others.

The rest of the paper is organized as follows. Section 2 defines the problem to be studied. A method of expressing position vectors with parameterized coordinates is introduced in Section 3. With this method, algebraic coupler curve was derived in Section 4. In Section 5, we extend the study of coupler curve to curvature analysis and path synthesis. Examples are included in Section 6. The work is discussed and concluded finally in Section 7.