In this paper, we present a solution-region-based synthesis approach for selecting optimal four-bar linkages with a Ball–Burmester point. We discuss both general and special cases of the Burmester point that coincide with the Ball point at the pole of the inflection circle. Given the coordinates of one fixed joint, any point on the target's straight line, and the direction of this straight line, we can synthesize an infinite number of mechanisms using a coupler curve with five-point contacts with its tangent by adopting the proposed approach. Each initial parameter corresponds to three side links that can generate three four-bar mechanisms. We generate different mechanism property charts by developing mechanism software that enables users to intuitively identify relevant linkage information and select the optimal linkage. This novel approach is a visualized analytical method for synthesizing and selecting optimal four-bar linkages with one Ball–Burmester point on its coupler curve.

As an important planar four-bar mechanism, four-bar linkages that approximate a straight line have been widely studied based on the theory of the kinematic geometry of mechanisms (Dijksman, 1976; Hunt, 1987; McCarthy, 2000; Wang and Wang, 2015). Dijksman (1972) and Dijksman and Smails (2000) used a geometrical method to discuss the Ball point in different configurations. Tesar et al. (1967) and Vidosic and Tesar (1967a, b) derived a series of synthesis formulas, and transformed the results into design diagrams for users according to three different cases, i.e., the general case of the Ball–Burmester point, the special case of the Ball–Burmester point at the inflection pole, and the special case of the Ball-Double Burmester point. Yu et al. (2013) presented a numerical comparison synthesis method for single and double straight-line guidance mechanisms to solve four-bar straight-line guidance mechanism synthesis problems for an arbitrarily given straight line's “angle requirement” and “point-position requirement”. Han (1993) studied the synthesis of the four-bar straight-line linkage of Ball and Burmester points in general cases. The author Yin et al. (2011, 2012) studied the synthesis of the straight-line linkage of Ball and Burmester points, separately. Han et al. (2009) proposed a solution-region-based method for linkage synthesis to obtain the optimal solution in the feasible solution region, and extended their approach to four-position finitely separated and mixed “point-order” positions (Yang et al., 2011), six-bar motion generation (Cui and Han, 2016), and RCCC Linkages (Han and Cao, 2018; Bai and Angeles, 2015). Traditional synthesis methods use congruence to represent infinite parametric solutions. The solution-region method is an optimal-mechanism synthesis approach for expressing infinite solutions on finite diagrams for cases irrespective of whether the congruence method can be used. Bai and Angeles (2015) proposed a new method for calculating cognate mechanisms, and cognate straight-line mechanisms can be obtained by employing this approach. However, none of the above authors have made a systemic study of how to choose desired straight-line mechanisms with a Ball–Burmester point from an infinite number of synthesized mechanisms, and that satisfy the target constraints.

Here, we present a visualized synthesis approach based on the solution region for selecting optimal four-bar linkages with a Ball–Burmester point. We discuss both the general and special cases of the Burmester point that coincide with the Ball point at the pole of the inflection circle. Different mechanism property charts are generated by developing mechanism software to enable users to intuitively identify information about the involved linkages and to select the optimal linkage from an infinite number of mechanism solutions.

Robert Ball proposed the famous Ball point theory, which is based on the
infinitesimal displacement and instantaneous invariance of curvature. The
Ball point is defined as the point whose radius of curvature is infinite and
whose curvature is stationary, which is the intersection point of the
inflection circle and the cubic of stationary curvature at a certain instant
(other than the polar point

The Burmester point is a higher-order stationary point of the cubic of a stationary curvature, whose trajectory intersects with the curve at no less than five infinitely close points, namely a four-order osculating. In this paper, we used the theory of the Burmester point to develop a method for synthesizing a five-point contact mechanism that approximates a straight line under both general and special conditions. The proposed method allows the designer to give a fixed hinge point, the points on the straight line, and the direction of the line.

Definitions of various parameters (Yin et al., 2012).

Assume a fixed joint point

According to the kinematic geometric theory of infinitely close positions,
several points can be selected from a motion plane at any instantaneous
position, whose trajectory has fourth-order contact with its curvature
circle. This means that these points are the circle points of five infinitely
close positions, namely Burmester points, which are higher-order stationary
points of the trajectory curvature. Using the Euler-Savary equation (Yin et
al., 2012) to solve the two-order derivative equation

After we determine the instantaneous center

According the relation between the fourth-order equation root and coefficient
of Eq. (1), we obtain:

we obtain the following:

There are two roots for

The values of

The coordinates of motion joints

Mechanism parameters.

Example diagrams of mechanisms.

When the Burmester point is located at the pole of the inflection circle,
parameter

Mechanism solution region diagrams for all mechanism solutions.

When the Burmester point coincides with the Ball point at the pole of the
inflection circle,

Synthesized mechanisms.

According to the given conditions, the solution region of the mechanism is
analyzed on the coordinate plane

The task is to design four-bar linkages that approximate a straight line
with the following conditions: fixed joint

There are three solution-region diagrams. In the mechanism-type solution
region diagram of mechanism

No solution,

Crank rocker,

Double rocker,

Rocker crank,

Double crank,

Triple rocker (in–out),

Triple rocker (out–out),

Triple rocker (out-in), and

Triple rocker (in-in).

Mechanism solutions with different displacements

Optimal mechanism and two cognate mechanisms.

Mechanism parameters.

If

To illustrate all the mechanism solutions in the finite coordinate plane,
displacement

When

If we let

In this task, we design four-bar linkages that approximate a straight line
and which satisfy the following conditions and requirements: Given a fixed
joint

Let

To synthesize the optimal mechanism solutions when the direction of the
straight line is 30

Let

Figure 6c shows that the relative length of the straight line is longest
when

In this paper, to select optimal four-bar straight-line linkages, we discussed both the general and special cases of the Burmester point that coincide with the Ball point at the pole of the inflection circle. By adopting our proposed approach, an infinite number of mechanisms with a coupler curve that has five-point contacts with its tangent were obtained. We generated different mechanism property charts by developing a mechanism software to enable users to intuitively identify information about the involved linkages and to select the one that is optimal. The results of the calculation examples indicated that the proposed method works effectively. This is a novel visualized analytical method for synthesizing and selecting optimal four-bar linkages with one Ball–Burmester point on its coupler curve.

Using the proposed method, we found there to be three straight-line linkages with the same straight-line segment of a coupler curve for each of the initial parameters given. However, the coupler curves of three cognate mechanisms by the Roberts–Chebyshev Theorem are identical. Therefore, after the initial parameters are given, we can synthesize three different mechanisms with the same straight-line segments of coupler curves by the proposed method. In addition, we can obtain the other two cognate mechanisms for each straight-line linkage based on the Roberts-Chebyshev Theorem.

All the data used in this manuscript can be obtained on request from the corresponding author.

LY proposed the idea and methodology; LH derived the equations; JH, LT and FL developed the software.

The authors declare that they have no conflict of interest.

This work has been financially supported by the National Natural Science Foundation of China (No. 51705034-51805047), the Natural Science Foundation of Hunan province (No. 2018JJ3548), and Innovation Platform Foundation of Key Laboratory of Safety Design and Reliability Technology for Engineering Vehicle of Hunan Provincial Department of Education (17K003). Edited by: Chin-Hsing Kuo Reviewed by: two anonymous referees