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It is often of engineering importance to design (synthesize) 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 positions of the coupler with a drawing board and with mathematics. Wee can also 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), center Point Curves, Burmester Points.
Definition of Position Synthesis:
Position Synthesis is 'find a mechanism to move a machine component through different, defined positions, relative to a different machine component.
We need a coordinate system to define the different positions. Luckily, Parts and Lines in MechDesigner have Coordinate Systems.
•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.
What is the Design Statement of Position Synthesis?
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. 
MechDesigner for Position Synthesis
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 position 

Step 1: We edit the BasePart to draw the two Lines at two positions. Each Line has an Origin (startPoint) and an Xaxis direction (direction from its startPoint to its endPoint). We can use the Line to represent a Plane relative to the axes of the BasePart. The two positions are:
Note: in the image, I have hidden the Point symbols. See Display toolbar > Show/Hide Points. 

Method 1 1.Select any two Points in the Plane at each position. Select the startPoint and the endPoint of each Line. 2.Draw a (red) construction line between equivalent Points in each Plane at each position  between the two startPoints and between the two endPoints.. 3.Add a (green) construction line that is perpendicular to the midpoint of each red line Key Learning 1 Consider the green construction line that we construct from the startPoints of each Line: A circle with its center anywhere on the green Line that also intersects the startPoint of one Line (Plane), will also intersect the startPoint of the other Line (Plane). Similarly, for the green construction line that we construct from the endPoints of each Line. 

4.Extend the two green construction lines until they meet at a coincident point Now, notice that you can: •Draw a circle at Point such that its circumference passes through Point in the two positions. •Draw a circle at Point such that its circumferences passes through Point in the the two positions. This means, we can use Point as the center of rotation for a Part that passes through Points and at the two Plane positions. We need to transfer from the PartEditor to add a mechanism in the MechanismEditor. 

Point can be the PinJoint of a Part that rotates. 1.We can now make various measurements in the MechanismEditor with the Measurement FB. 2.Measure the Radius of the Circle that passes trough one of the Points in the Plane  R50.5mm 3.Measure the angle from the horizontal and the Line at the first position of the Plane  120.0º 4.Add Linesfrom the center of the circle to the Point(1) in both positions. 5.Measure the angle between the two Lines  40º 6.Measure the angle between Line and a Line that joins the two Points in the Plane  130.0º 

You have all the information to: 1.Add a Part 2.Join it with a PinJoint and add a MotionDimension FB  edit it to enter the correct BaseValue  120.0º. 3.Add a sketch to the Part to represent the Plane  solid Green 4.Rotate the Part with a Motion of 40º 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.

It is not possible to make a circle coincident with four(4) arbitrary(random) Points on a Plane. You can make a circle pass through 3 arbitrary points, but not four. To guide a point through 4 Positions, we need to find a special Point within in the Plane of the Coupler, that does pass through a circle in the Frame. In the PartEditor 1.Add four Lines. 2.Add Equal constraints to make the Lines all the same length. 3.Fix the Points  use the Point Vector and Position dialogbox. These are the four separate position of the Plane. They identify the origin and orientation of the Coupler Plane. Labeling As you know, each Line has an Origin and an Xdirection. I have labeled the Origins as 01, 02, 03, 04. I have labeled the Points along the Xaxis as X1, X2, X3, X4. 

4.Add two Lines to make a triangle with the fixed Lines 0n, Xn. 5.Label the apex of the triangle A1, A2, A3, A4. 6.Add Equal constraints to the four lines, An0n, and Equal Constraints to the four lines An Xn to make the triangles congruent. 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 should remain congruent. 

Design Objective As you know a circle can be made to pass through THREE Points. A Circle cannot be made to pass through FOUR Points. We hope to find a circle that passes though Points A1, A2, A3 and A4. MechDesigner must search for a special Point within the Plane of the Coupler that does pass through one circle. 7.Add a Circle 8.Add Coincident Constraints between the Circle and the Apex of the four triangles, A1 to A4. 9.Label the center of the Circle A0 Step 8 is complete. 

10.Do Steps 4 to 6 for a different Triangle. Label the apex of the triangles B1, B2, B3, B4. 11.Do Steps 7 and 8 again for B1 to B4 12.Label the centers of the Circle B0 Step 12 is complete 

Add Parts in the MechanismEditor 1.Add a Part and Join it with a PinJoint to center of a Circle 2.Add a different Part and Join it with a PinJoint to the center of the other Circle 3.Make the Parts the same length as radii of the Circles 4.Measure the distance between the Points A1 and B1  the length of the Coupler 5.Add a Coupler and make it the correct length 6.Join it to the end of the Parts 7.Add a MotionDimension FB to move it through the arc 


You can edit the diameters of the circles to find other positions for the centers of the circles, A0 and B0, and the corresponding Points for An and Bn. The images to the left show different centers for the circles. This gives: •different positions for the PinJoints in the frame •different lengths for the Parts •different positions for the PinJoints in the Coupler Part You can experiment. Different results may improve the results. For example the transmission angle of the Parts may be improved. It is much more likely that there is a defects  for example, the mechanism will move from 1 to 2 to 4 and mechanism must be in a different closure to move through 3.

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. 