Four Bar Motion Synthesis

This is part of a series for synthesizing four bar linkages using MEboost.  In this article we will use four bar motion synthesis to design linkages.  It's recommended to view part 1 for terms and conventions used in MEboost.

Part 1: Four Bar Linkage Synthesis

Part 2: Four Bar Path Synthesis

Part 3: Four Bar Function Synthesis

Four Bar Motion Synthesis

With four bar motion synthesis, we are not only concerned that a point on a link will follow the path we specify, but the link with the point also is at a specific angle at each precision point.  The angle of the input link (crank) may or may not be of concern.  When the crank angle is not important, it's called untimed.  For timed synthesis, the crank angle must be at a given angle when the point is at a specific precision point.

Because of the requirements of motion synthesis, a four bar linkage may not be able to perform the required motion.  It's advisable to use the bare minimum of precision points to allow for a possible solution.


With motion there are two types of error: point and angle.  These errors are shown below.  Maximum point and angle errors are the largest point and angle error.

Motion is a multi-objective optimization problem.  We want to minimize both the point error and the angle error.  MEboost uses compromise programming to determine a composite objective function.  The error below is minimized during the synthesis process.

There are optional weighting factors for point and angle errors.  These can be used for two purposes.

  1. To give more importance to either point or angle error.  For example, if point error is more important, a higher point weight can be assigned to force the algorithm to lower point error at the expense of angle error.
  2. In some instances, one of the objectives is easier to find.  Weights can be used to balance the errors to find a better overall solution.  For example, if angle error quickly goes to a small value while point error does not converge very well, assigning a high point weight of say .95, and a low angle weight of say .05 will put more focus on reducing point error.  Weights can be of any value, except they both cannot be zero.

If point and angle weights are not entered, it is assumed that each weight = 1.  It's recommended not to use weights on the first run to see how it is working.

Motion Synthesis Examples

We'll do an example of untimed and timed synthesis.

Untimed Example

The precision points/linkage angles and the plot of these points are shown below.  The coupler containing the point is link 3.  Therefore we want to control the angle of link 3 at each precision point.

Open the four bar synthesis form.

Synthesis Type

This will be motion synthesis so select motion.

Link of Interest

The link of interest is the link that has the point that will follow the path and the angles it must be at for each precision point.  We will use link 3 which is the coupler.


Motion is where we define the precision points and link angles.  The precision point coordinates and link angles must be entered in a worksheet.  You can type the range address directly in the box or select the range.

To select link 3 angle, click the minimize button to the right of the link 3 angles box.  A selector window will appear.  Select the range containing the angles and click OK.

To select the x coordinates, click the minimize button to the right of the path x values box.  A selector window will appear.  Select the range containing the coordinates and click OK.

Do the same for the y coordinates.


We can constrain the dimensions of the linkage by entering minimums and/or maximums in the appropriate boxes.  For this example, we won't add any constraints.


We need to set a stopping criteria so that when error drops to this level the synthesis process will stop.  You can also stop the process manually as we'll do later.


When the synthesis process stops, a report will be created for the results.  You can create the report in a new sheet in the current workbook or in a new workbook.

Now we're ready to start four bar motion synthesis.  Click run and a progress form will appear.  The progress form will show average point error, average angle error, and the current best solution.  NOTE: The process is paused to capture this screen shot.  To resume, click the Run button on the progress form.

Once errors reach an acceptable level, we can manually stop the synthesis process by clicking the Stop button on the progress form.  The synthesis report will then be created.  The report shows the synthesized linkage dimensions, a comparison of precision points with the linkage point path, a comparison of precision point link angles with synthesized link angles, and an error log.

The point and link angles compared to the precision points is magnified below.

The final errors are:

Average point error = .178

Average angle error = .027

Timed Example

The precision points/linkage angles for a path and the plot of these points are shown below.  Since this is a timed example, link 2 angles are included.

Open the synthesis form, select motion for synthesis type and select link 3 for link of interest.  Since this is a timed example, click on the Timed positions radio button in the Motion frame.

When Timed positions is selected, a box for the link 2 angles will appear.  Select the range of values for link 2 angles.

Select the link 3 angles, x and y coordinates the same as we did in the untimed example.

In this example, we'll use maximum error.  The results are shown below.

The point and link angles compared to the precision points is magnified below.

The final errors are:

Maximum point error = .457

Maximum angle error = 4.154