It is often of engineering importance to design (synthesise) a fourbar mechanism that guides a machine component, often a tool or endeffector, through a number of positions.
When the machine component is the coupler of a fourbar, the synthesis is frequently called Position Synthesis.
It is possible to find the postions of the coupler with drawing aids on a drawing board, and by mathematics. However, we can obtain wonderful insights of the design process when we use ConstraintBased Sketch Tools, especially when we want to find four or five separate positions. Also, the tools give us insights of Circle Point Curves (Cubic of Stationary Curvature), Centre Point Curves, Burmester Points.
Definition of 'Position':
'Position Synthesis' is 'find a mechanism to move a machine component to different, defined positions that are relative to another machine component.
We need a coordinate system in each Part with which we can define their positions. Luckily all Parts and Lines in MechDesigner have a Coordinate System.
•  Part's Coordinate System: Origin is its 'startPoint'; Xaxis is along its CADLine towards its endPoint. 
•  Line's Coordinate System: Origin is its startPoint; Xaxis is along it towards the endPoint. 
Thus, we can use Lines in the PartEditor in the same way as we use Parts in the MechanismEditor.
We use geometric constraints in the PartEditor and Joints in the MechanismEditor.
This is because:
•  'CoincidentConstraint' between two Points is exactly the same as the 'PinJoint'. 
•  'CoincidentConstraint' between two Lines is exactly the same as the 'SlideJoint'. 
Parts and Lines have an 'Origin' and 'Xaxis' direction. Therefore, we can represent a 'Plane' and its Position with a Part in the MechanismEditor and a Line in the PartEditor.
Design Input: Guide a Plane or Part through a number of specified positions. Design Output: A fourbar mechanism that guides its coupler through each specified position. 
Two and Three Position Synthesis: We can do the synthesis for two and three position synthesis in three ways: Procedure 1: Uses traditional graphical techniques ('use a compass to draw arcs, perpendicular bisectors, intersections...') Procedure 2: Uses 'traditional graphical techniques' and also simple constraints for the arcs and lines. Procedure 3: Uses only the ConstraintBased Sketch Tools There is an example of each method in the ThreePosition Synthesis. Four and Five Position Synthesis: Procedure 4: Uses more Advanced use of the ConstraintBased Sketch Tools Rather than find a solution for cubic curves, we use the powerful ConstraintBased Sketch Tools in the PartEditor to find two Points in the coupler Part that move around the circumference of a circle. Note: It is not easy to find a solution for five different positions. Small changes to the Plane Positions can give kinematic results that are amazingly different. You might need to experiment with the four and five position synthesis for a long time to get a satisfactory result. 
To guide a Plane from position to a different position, at position 

Step 1: We use the PartEditor to draw the two Lines at the two positions. Because each Line has an Origin and an Xaxis direction, we can use it to represent a Plane. The two positions are:


Method 1
In this case, it is easy to select the startPoint and the endPoint of each Line.
Key Learning 1 When you draw a circle, with its centre anywhere on the green construction Line that relates to Point, and it passes through Point in the first Plane position it will also pass through Point in the second position. Similarly for the construction given for Point 2. 

Now, notice that you can:
This means, we can use Point as the centre of rotation for a Part that passes through Points and at the two Plane positions. We need to transfer the learnings from the PartEditor to add a mechanism in the MechanismEditor. 

Point can be the PinJoint of a Part that rotates.


You have all the information to:
You will see the Green Plane move between the two positions. The two position synthesis is complete. 
We can show three graphical construction procedures.
The first procedure is exactly as we would if we had a piece of paper, a compass to draw arcs and circles, and a rule to draw straight lines.
The second procedure uses a few constraints. Procedure three uses constraint to the full, and is much faster.
There is of course a Mathematical procedure we can follow.
Position of Plane n 
Plane Origin X(mm) 
Plane Origin Y(mm) 
Θ(degrees) of Xaxis/Line/Plane 

Position 1 
3 
53 
90 
Position 2 
31 
59 
60 
Position 3 
60 
50 
0 
In 3PS, you have two Points in a Plane, Point A and Point B. They move through three Positions, 1, 2 and 3. The key is to find the intersection of two perpendicular bisectors. In the PartEditor Sketch Lines as the three Positions of the Part in the MechanismPlane
I have labelled the Points in the image as: A1,B1, A2,B2, A3,B1. 

In the PartEditor
Draw the Arcs so that they intersect at two places.
This is a Perpendicular Bisector.
A0 is a common rotation point for A. An arc with a centre at Point A0, with a radius A0A, passes through the Positions A1, A2 and A3. 

The construction is over. You can draw arcs in the PartEditor.


In the MechanismEditor


The fourbar mechanism (the BasePart is one Part, if you did not know) moves the coupler AB, defined by Point A and B, through the Positions 1, 2 and 3. 
This procedure is only slight better than procedure 2. It uses the same 'Perpendicular Bisector' ideas. However, we introduce a few more Constraint Tools. 

In the PartEditor Sketch Lines as the three Positions for the Part in the MechanismPlane
I have labelled the Points in the image as: A1,B1, A2,B2, A3,B1. 

Add the Geometric Construction to find the Centres and the Lengths of the Parts.
(Add a Line between Point A1 and A2. (A1A2) Add a different Line. Constrain one end to the midPoint of A1A2, and also make it Perpendicular to A1A2)
If you put a compass point at A0, you can draw an arc from A1 to A3. It will also pass through A2. If you put a compass point at B0, you can draw an arc from B1 to B3. It will also pass through B2. 

In the MechanismEditor
Use Measurement FBs. Note shown in image.
(I have added Blue lines to show the fourbar mechanism). 

Notes: If you choose other Points in the Part, you will get a different mechanism. You can choose all possible Points in the Plane for A and all possible Points in the Plane for B. Somehow, that makes an infinitysquared number of choices. 
Procedure 3: Modern Constraint Based  three Points on an Arc You find a circle  its radius and centre  that intersects with Points A and B at Positions 1,2 and 3. You can always find a circle that passes through three Points. You use the constraint based sketch editor and add coincident constraints between a circle and Points A1, A2, A3. You add a different circle for Point B at Positions 1,2 and 3. You can use the MechanismEditor to add the Parts with the correct lengths. 

This procedure is much easier and much quicker. It uses the Constraint Based Sketch Editor In the PartEditor Sketch Lines as the three Positions for the Part/Plane in the MechanismPlane
You can label the Points: A1, B1, A2, B2, A3, B1. 

Add circles and Coincident Constraints


In the MechanismEditor
Use Measurement FBs. Note shown in image.
(I have added Blue Lines to show the fourpart mechanism). 


In the PartEditor
These are the four separate position of the Plane. They identify the origin and orientation of the Coupler Plane. Labelling As you know, each Line has an Origin and an Xdirection. I have labelled the Origins as 01, 02, 03, 04. I have labelled the Points along the Xaxis as X1, X2, X3, X4. 

Make sure the triangle are in the same quadrant relative to the Line 0n,Xn Step 6 is complete If you drag the apex of one triangle, all of the triangles move together as they remain congruent. 

Design Objective We hope to find a circle that passes though Points A1, A2, A3 and A4. The Points, An, move within the Coupler Plane as MechDesigner searches for a solution.
Step 8 is complete. 

Label the apex of the triangles B1, B2, B3, B4.
Step 12 is complete 

Add Parts in the MechanismEditor



You can edit the diameters of the circles to find other positions for the centres of the circles, A0 and B0, and the corresponding Points for An and Bn. The images to the left show different centres for the circles. This gives:
You can experiment. Different results may be better. For example the transmission angle of the Parts may be improved. 

FIVE POSITION SYNTHESIS IS VERY SIMILAR. YOU MUST MOVE THE CIRCLES UNTIL A FIFTH POINT ALIGNS WITH THE CIRCLES... 
Make a Table of Position that you would like the Coupler to Guide a Plane through 

The Planes are defined by the five lines The Far image shows that I have made four Points at the apex of the congruent triangles coincident to the circle. The Near image shows that all five Points at the apex of the congruent triangles are coincident to the Circle. The sketch is black to indicate that I cannot add more constraints. 

The Far image shows I have added five more congruent triangle to the Plane Positions The Near image shows the five Points at the apex of the triangles are coincident with a different circle. 

The Far image shows the same construction, but in the MechanismEditor. The near images shows the fourbar mechanism chain. The coupler part has been extended so it moves between and to the five positions that are defined by the Lines in the PartEditor. 



Make a Table of Position that you would like the Coupler to Guide a Plane through 

The Planes are defined by the five lines. I have added five pair of lines to each Plane to make a triangle. I have added Equal Constraints to the similar sides to make the triangle to be 'congruent triangles'. The near image shows that I have made four Points at the apex of the congruent triangles coincident to the circle.


The Far image shows that all five Points at the apex of the congruent triangles are coincident to the Circle. The sketch is black to indicate that I cannot add more constraints. The Far image shows the same construction, indicating five planes moving through the five positions.


The Far image shows I have added five more congruent triangle to the Plane Positions The Far image shows the five Points at the apex of the triangles are coincident with a different circle. The near images shows the fourbar mechanism chain. 