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# Planar Linkage Synthesis for Mixed Motion, Path, and Function Generation Using Poles and Rotation Angles1OPEN ACCESS

[+] Author and Article Information
Ronald A. Zimmerman, II

Mem. ASME
Product Engineering Specialist
Research and Development,
Magna Seating,
Troy, MI 48098
e-mail: ron.zimmerman@magna.com

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 13, 2017; final manuscript received January 2, 2018; published online February 5, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 10(2), 025004 (Feb 05, 2018) (8 pages) Paper No: JMR-17-1294; doi: 10.1115/1.4039064 History: Received September 13, 2017; Revised January 02, 2018

## Abstract

The kinematic synthesis of planar linkage mechanisms has traditionally been broken into the categories of motion, path, and function generation. Each of these categories of problems has been solved separately. Many problems in engineering practice require some combination of these problem types. For example, a problem requiring coupler points and/or poses in addition to specific input and/or output link angles that correspond to those positions. A limited amount of published work has addressed some specific underconstrained combinations of these problems. This paper presents a general graphical method for the synthesis of a four bar linkage to satisfy any combination of these exact synthesis problems that is not overconstrained. The approach is to consider the constraints imposed by the target positions on the linkage through the poles and rotation angles. These pole and rotation angle constraints (PRCs) are necessary and sufficient conditions to meet the target positions. After the constraints are made, free choices which may remain can be explored by simply dragging a fixed pivot, a moving pivot, or a pole in the plane. The designer can thus investigate the family of available solutions before making the selection of free choices to satisfy other criteria. The fully constrained combinations for a four bar linkage are given and sample problems are solved for several of them.

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## Introduction

In linkage synthesis, there are three traditional tasks: motion, path, and function generation. In motion generation, the poses or positions and orientations of a coupler link are specified. Path generation specifies only the positions of a coupler point. Function generation coordinates the angles of the input and output links.

Generally, each of these problems is treated separately. However, there are cases when a mechanism needs to perform some combination of these tasks. For example, a coupler may need to pass through several points and/or poses, while the input and/or output links have specified orientations at some coupler positions. I call these mixed synthesis problems. Hall [1] gives an example of a walking mechanism that falls into this category. Stowing automotive seats frequently also require a combination of motion and function generation. Collapsing mechanisms are often primarily motion generation problems, but in the collapsed position, there are frequently orientation requirements on the input links so they will fit within the design space. A foot pedal actuator for a drum requires coupler link positions to be coordinated with input link angles [2]. Solutions for some underconstrained mixed synthesis cases are known. Erdman and Sandor show a three point, two input angle and a three pose, two input angle solution method [3]. Beyer shows a graphical method for two coupler points and specified input and output angle changes corresponding to those two points [4]. A general procedure for solving any combination of exact motion, path, and function generation tasks is lacking. Recently, a solution to the mixed motion and path generation problem was published [5,6]. I am unaware of any attempt to consider the fully constrained cases of the mixed motion, path, and function synthesis problem.

New tools often give rise to new discoveries. The recent advent of constraint based sketchers in modern cad software is a new tool. This has opened the door to new ways of designing planar linkages [7,8]. One of these tools is pole and rotation angle constraints (PRCs) [9]. cad systems are widely used in academia and industry. cad sketchers are easily the most prevalent tool used for mechanism design. Most mechanism design problems are also underconstrained. Mechanism design methods that can be understood by practicing engineers, are easily applied to underconstrained problems, and are easily implemented in cad are highly desirable. Pole and rotation angle constraints fit this need. It is a powerful, intuitive graphical method utilizing modern cad sketchers. It can be used to solve any planar mechanism design problem that is not overconstrained. The more underconstrained the problem is, the easier PRC is to apply. This and the simplicity of PRC make it an ideal method for the practicing engineer to learn and use.

Pole and rotation angle constraints are based on the geometric constraints imposed by the target positions on the linkage through the poles and rotation angles. Pole and rotation angle constraints have been shown to be a useful tool for solving the motion, path, and function generation problems separately [911]. In this paper, PRC is extended to solve the fully constrained and any underconstrained mixed synthesis problems. Since PRCs are necessary and sufficient conditions for hitting the target positions, if the problem is underconstrained, the available solutions can be observed by dragging fixed or moving pivots or a pole in the plane before making any free choices. The free choices can be used to optimize important features of the mechanism. If the solution is underconstrained by one, the solution curve is discontinuous. Only the segment you create the constraints on can be investigated.

Consideration of the function generator problem in this paper is restricted to making the input or output links have a given orientation at a specified coupler position. For example, a guiding link rotates through some angle θ, while the coupler moves from positions 1 to 3 as shown in Fig. 1 or the output link is at a specified angle relative to a ground reference frame at a given coupler position as illustrated in position 4 of the example in Fig. 2 and Table 1. My experience is that this is the most likely practical need. However, similar to a traditional function generator problem, it is possible to consider function generation independent of coupler positions. For example, the output link rotates through an angle ψ, while the input link rotates θ independent of the coupler positions as illustrated in Fig. 3. This case is not included here but is a topic for future consideration. Developing the equations for an analytical solution to the mixed synthesis problem is also a topic for future research.

Each mixed synthesis problem combination is labeled according to the following convention: M-P-F (motion, path, function), based on how many of each type of target positions there are. For example, a problem with two target coupler poses (M), three target coupler points (P), and one target input angle change (F) would be classified as 2-3-1. The cad program Catia V5-6R2014™ was used in all the examples throughout the paper. Consideration is limited to four bar linkages.

## Mixed Synthesis

Many mechanism design problems require a combination of motion generation, path generation, and/or function generation. These can be termed mixed synthesis problems. The procedure for solving mixed synthesis problems is to use the basic pole and rotation angle constraints required for each piece of the problem and superimposing them.

As a review, the basic constraint for two motion generation poses is given in Fig. 4. For any two positions, the constraint is that the fixed pivot and the moving pivot are on lines going through the pole which are separated by half the rotation angle (Ø). The rotation from the moving pivot line to the fixed pivot line is in the same direction as the rotation angle. The basic constraint for two coupler points in the path generation problem is given in Fig. 5. For any two points, the constraint is that the pole must be on the perpendicular bisector of the two points. In addition, the coupler rotation angle is dependent on the location of the pole along this line, and the fixed and moving pivots are on lines passing through the pole which are separated by half the coupler rotation angle. The rotation from the moving pivot line to the fixed pivot line is in the same direction as the rotation angle. The fixed and moving pivots can be located anywhere on the fixed and moving pivot lines. When this pair of lines is rotated around the pole, it represents the family of all the solutions to the two position problem.

The basic constraint for function generator target positions is given in Fig. 3. Since the input link angles correspond to coupler positions, the fixed and moving pivot lines have already been created for the coupler positions using the motion or path generation procedures just mentioned. The desired angular constraint is then applied to the guiding link. This can be an angular constraint relative to a fixed reference or an angular change between two positions. In either case, the fixed pivot line of the target coupler positions is then constrained to bisect the corresponding positions of the guiding link.

These basic constraints are added for each respective coupler pose, point, and guiding link angle. At this point from each pole, there will be two fixed and moving pivot line pairs. One pair for each guiding link to be found. Each pair of lines is free to rotate around its pole and each pivot can be located anywhere on its line. Next, place a point on a moving pivot line. Then, constrain all the other moving pivot lines for that guiding link to pass through that point. Repeat the same procedure for the corresponding fixed pivot lines and then the fixed and moving pivot lines of the other guiding link. Pivots that will satisfy multiple positions are found at the intersections of their respective pivot lines, i.e., the places where the fixed pivots lines from P12, P13, and P14 intersect will satisfy positions 1–4. When all of the constraints for both guiding links are made, a solution mechanism is found. If the problem is underconstrained, the designer will then be able to drag the fixed or moving pivots or a pole around in the plane until the most desirable solution is identified.

###### Mixed Synthesis Illustration.

A sample problem is given in Fig. 1. The coupler must pass through two target poses and one target point. The input link must rotate through a specified angle as the coupler goes from pose 1 to 3. This is a 2-1-1 case.

Beginning with coupler pose one, three individual constraints must be applied. One for coupler point 2, one for coupler pose 3, and one for the input link rotation between coupler poses 1 and 3. The three constraints are shown in Fig. 6.

Step 1: Apply constraints for the motion generation problem of positions 1 and 3. Find pole P13 and rotation angle Ø13. Draw two sets of fixed and moving pivot lines from the pole. One for the input link and one for the output link. Step 2: Apply the function generator requirement that the input link rotates through the angle θ13, while the coupler moves from positions 1 to 3. The angle between the input link in position one and the fixed pivot line from P13 is ½θ13. Step 3: Apply the point path constraint for coupler point 2. Find the line of P12. Measure the rotation angle Ø12 for a given P12 location. Note that Ø12 is a variable. It varies with the distance of P12 from the target points. Draw two sets of fixed and moving pivot lines from P12, one for the input link and one for the output link. Step 4: Constrain the fixed and moving pivots of the input and output links to be at the intersection of their respective fixed and moving pivot lines. Steps 1–3 can be done in any order.

Since this problem is underconstrained, the guiding links can be dragged around in the plane until optimal pivot locations are found. This procedure of successively adding motion, path, and function generator constraints can be applied to any mixed synthesis problem that is not overconstrained.

The solutions given in this paper have revolute joints. However, this method will also find prismatic joint solutions. When the designer is dragging a pivot around in the plane to find an optimal solution, there are times when the fixed or moving pivot lines become parallel and the pivot goes to infinity. These are locations of prismatic joints. A solution to this sample problem with a prismatic joint is shown in Fig. 7. The prismatic joint is between the coupler link and the guiding link with a fixed pivot at O2. These two links have a constant orientation with respect to each other. In position 1, the line of sliding is perpendicular to the moving pivot lines.

An observation about function generator synthesis methods bears noting. When discussing the function generator problem, most references observe that it can be solved like a motion generation problem if inverted [12,13]. However, this approach is not easily applied to mixed synthesis problems. In contrast, the direct function generator solution shown earlier [11] can be applied to the mixed synthesis problem. This illustrates a distinct advantage of solving the function generator problem directly as opposed to by inversion.

For any planar synthesis problem that is not overconstrained, the designer simply has to ask “Where are the poles, and what are the rotation angles?” followed by the application of one rule; viz., the fixed and moving pivots are on lines through the pole separated by half the rotation angle. Pole and rotation angle constraints do not preclude branch or order defects. It simply guarantees that the mechanism can be assembled in the target positions. The synthesized mechanism needs to be checked for these conditions.

###### Fully Constrained Mixed Synthesis Problems.

One of the first questions that comes to mind is to determine what constitutes a fully constrained problem, i.e., what combinations of coupler poses, coupler points, and driving link angles is the maximum capable of being reached by a four bar linkage.

If M represents the number of motion generation poses, P the number of path generation points, and F the number of function generator angles, then the fully constrained design limit is any combination for which Display Formula

(1)$10=2M+P+F$

In this equation, M cannot be zero. As noted by Brake et al. [5] regarding mixed motion and path generation problems, M = 0 and M = 1 are equivalent conditions. If no motion poses are specified, then an orientation requirement can be added to any one specified point without affecting the solution. All fully constrained combinations are given in Table 2. A function generator target angle means one specified angle of either guiding link. Specifying a travel angle for both the input and output links at a given coupler position counts as two constraints, i.e., F = 2. In addition, since function generator constraints must correspond with a specified coupler link pose or point then Display Formula

(2)$F≤2(M+P−1)$
i.e., there cannot be more than two guiding link angles specified for any coupler position. This eliminates the dot hatched entries in Table 2. Combinations for which Display Formula
(3)$10>2M+P+F$
are underconstrained and can also be solved.

I have documented solutions to sample problems for all the combinations colored gray which account for 15 of the 20 possible combinations. Several examples follow.

The target coupler points and poses can be numbered one through nine. The input or output link target angles are labeled based on the coupler position they match. Each mixed synthesis problem combination is labeled according to the following convention. M-P-F based on how many of each type of target positions there are. The sample problem in Sec. 2.1 is 2-1-1, meaning the targets are two motion poses, one coupler point, and one function generation angle. There are multiple possible configurations of targets within most combinations. For example, within the 3-2-2 combination, the three coupler poses and two points can be in any order and the two input link angle requirements can both be on one link, or one on each link and can correspond to any of the coupler poses or points. Two different 3-2-2 configurations (out over a hundred possible permutations) are shown in Fig. 8.

It is also noted that the individual motion, path, and function generation problems are special cases of the general mixed synthesis problem. In Table 2, the motion generation problem is represented by the case 5-0-0, the path generation problem is represented by the case 1-8-0 and the traditional function generator problem is represented by the case 1-0-8.2

###### Fully Constrained Mixed Synthesis Solutions.

Target positions and solution mechanisms are provided for four fully constrained cases in Figs. 2 and 911 and Tables 1 and 35. A designer who is competent at using a cad sketcher and understands how to make pole and rotation angle constraints can usually find a solution to any of these fully constrained problems from scratch in 30–45 min assuming a reasonable set of target positions for which a solution can be expected.

The fully constrained 2-4-2 case shown in Figs. 10, 12, and 13 has two target poses, four target points and two target input angles. The first step is to locate the poles and identify the rotation angles as shown in Fig. 12. Doing the synthesis in position 1, there are five poles, P12, P13, P14, P15, and P16. Two sets of fixed and moving pivot lines are drawn from each pole, one for each guiding link. Fixed and moving pivots are located at the intersection of five fixed and moving pivot lines, respectively, one from each pole. See Fig. 13. The same procedure is followed for the other problems.

Fully constrained cases are more challenging to solve than underconstrained cases. Often when making the last constraint, the range of values available in this final degree-of-freedom is limited. The Burmester curves of the four position motion generation problem sometimes contain solution segments that are not connected. In this case, when the constraint for the fourth position is made on one solution segment, the other separate segment cannot be observed by dragging a pivot around the plane. The other segment can be observed only if the last constraint is removed and remade on it. Similarly, the solution set for mixed synthesis problems underconstrained by one as given in Eq. (4) seems to also be frequently discontinuous. Analogously for the fully constrained mixed synthesis problem, only the segment of the solution curve on which the penultimate constraint is made can be searched by dragging to find solutions for the final target position. The solution(s) for the final position may lie on another segment of this curve in which case a constraint would have to be removed and remade Display Formula

(4)$9=2M+P+F$

## Conclusion

The use of PRC provides a method for the exact synthesis of mixed motion, point, and function generation four bar linkage problems. The designer simply has to ask Where are the poles, and what are the rotation angles? followed by the application of one rule; viz., the fixed and moving pivots are on lines through the pole separated by half the rotation angle.

Pole and rotation angle constraints is an ideal method for solving planar mechanism design problems. It utilizes the most ubiquitous design tool (cad). It is visual, intuitive, and simple to learn and use. It is applicable to any design problem that is not overconstrained. The more underconstrained the problem is, the easier it is to apply. These attributes make PRC an ideal method for the practicing engineer and student to learn and use.

## Acknowledgements

Magna Seating provided the tools and support needed to conduct this research.

## References

Hall, A. S., Jr ., 1986, Kinematics and Linkage Design, Waveland Press, Prospect Heights, IL, p. 8.
Erdman, A. , and Sandor, G. , 1991, Mechanism Design Analysis and Synthesis Volume 1, Prentice Hall, Englewood Cliffs, NJ, pp. 7 and 37.
Erdman, A. , and Sandor, G. , 1991, Mechanism Design Analysis and Synthesis Volume 1, Prentice Hall, Englewood Cliffs, NJ, pp. 535–551.
Beyer, R. , 1963, The Kinematic Synthesis of Mechanisms, McGraw-Hill, New York, pp. 178–182.
Brake, D. , Hauenstein, J. , Murray, A. , Myska, D. , and Wampler, C. , 2016, “ The Complete Solution of Alt-Burmester Synthesis Problems for Four-Bar Linkages,” ASME J. Mech. Rob., 8(4), p. 041018.
Tong, Y. , Myska, D. , and Murray, A. , 2013, “ Four-Bar Linkage Synthesis for a Combination of Motion and Path-Point Generation,” ASME Paper No. DETC2013-12969.
Kinzel, E. C. , Schmiedeler, J. P. , and Pennock, G. R. , 2006, “ Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming,” ASME J. Mech. Des., 128(5), pp. 1070–1079.
Mirth, J. , 2012, “ Parametric Modeling—A New Paradigm for Mechanisms Education?,” ASME Paper No. DETC2012-70175.
Zimmerman, R. , 2013, “ Planar Linkage Synthesis for Rigid Body Guidance Using Poles and Rotation Angles,” ASME Paper No. DETC2013-12036.
Zimmerman, R. , 2014, “ Planar Linkage Synthesis for Coupler Point Path Guidance Using Poles and Rotation Angles,” ASME Paper No. DETC2014-34058.
Zimmerman, R. , 2015, “ Planar Linkage Synthesis for Function Generation Using Poles and Rotation Angles,” ASME Paper No. DETC2015-46240.
Uicker, J. , Pennock, G. , and Shigley, J. , 2011, Theory of Machines and Mechanisms, Oxford University Press, New York, pp. 441–445.
McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages, 2nd ed., Springer, New York, pp. 119–120.
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## References

Hall, A. S., Jr ., 1986, Kinematics and Linkage Design, Waveland Press, Prospect Heights, IL, p. 8.
Erdman, A. , and Sandor, G. , 1991, Mechanism Design Analysis and Synthesis Volume 1, Prentice Hall, Englewood Cliffs, NJ, pp. 7 and 37.
Erdman, A. , and Sandor, G. , 1991, Mechanism Design Analysis and Synthesis Volume 1, Prentice Hall, Englewood Cliffs, NJ, pp. 535–551.
Beyer, R. , 1963, The Kinematic Synthesis of Mechanisms, McGraw-Hill, New York, pp. 178–182.
Brake, D. , Hauenstein, J. , Murray, A. , Myska, D. , and Wampler, C. , 2016, “ The Complete Solution of Alt-Burmester Synthesis Problems for Four-Bar Linkages,” ASME J. Mech. Rob., 8(4), p. 041018.
Tong, Y. , Myska, D. , and Murray, A. , 2013, “ Four-Bar Linkage Synthesis for a Combination of Motion and Path-Point Generation,” ASME Paper No. DETC2013-12969.
Kinzel, E. C. , Schmiedeler, J. P. , and Pennock, G. R. , 2006, “ Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming,” ASME J. Mech. Des., 128(5), pp. 1070–1079.
Mirth, J. , 2012, “ Parametric Modeling—A New Paradigm for Mechanisms Education?,” ASME Paper No. DETC2012-70175.
Zimmerman, R. , 2013, “ Planar Linkage Synthesis for Rigid Body Guidance Using Poles and Rotation Angles,” ASME Paper No. DETC2013-12036.
Zimmerman, R. , 2014, “ Planar Linkage Synthesis for Coupler Point Path Guidance Using Poles and Rotation Angles,” ASME Paper No. DETC2014-34058.
Zimmerman, R. , 2015, “ Planar Linkage Synthesis for Function Generation Using Poles and Rotation Angles,” ASME Paper No. DETC2015-46240.
Uicker, J. , Pennock, G. , and Shigley, J. , 2011, Theory of Machines and Mechanisms, Oxford University Press, New York, pp. 441–445.
McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages, 2nd ed., Springer, New York, pp. 119–120.

## Figures

Fig. 1

Mixed synthesis case 2-1-1

Fig. 2

Case 3-2-2 targets and solution

Fig. 3

Function generation constraint

Fig. 4

Motion generation constraint

Fig. 5

Path generation constraint

Fig. 6

Mixed synthesis sample constraints

Fig. 7

Solution with prismatic joint

Fig. 8

Two 3-2-2 configurations

Fig. 9

Case 1-4-4 targets and solution

Fig. 10

Case 2-4-2 targets and solution

Fig. 11

Case 4-1-1 targets and solution

Fig. 12

Case 2-4-2 pole and rotation angle constraints

Fig. 13

Case 2-4-2 fixed and moving pivot lines intersections

## Tables

Table 1 Case 3-2-2 targets and solution
Table 2 Fully constrained combinations
Table 3 Case 1-4-4 targets and solution
Table 4 Case 2-4-2 targets and solution
Table 5 Case 4-1-1 targets and solution

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