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Two, Three, Four and Five Position Synthesis

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Two, Three, Four and Five Position Synthesis

It is often of engineering importance to design (synthesise) a four-bar mechanism that guides a machine component, often a tool or end-effector, through a number of positions.

When the machine component is the coupler of a four-bar, 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 Constraint-Based 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 'start-Point'; X-axis is along its CAD-Line towards its end-Point.

•	Line's Coordinate System: Origin is its start-Point; X-axis is along it towards the end-Point.

Thus, we can use Lines in the Part-Editor in the same way as we use Parts in the Mechanism-Editor.

We use geometric constraints in the Part-Editor and Joints in the Mechanism-Editor.

This is because:

•	'Coincident-Constraint' between two Points is exactly the same as the 'Pin-Joint'.

•	'Coincident-Constraint' between two Lines is exactly the same as the 'Slide-Joint'.

Parts and Lines have an 'Origin' and 'X-axis' direction. Therefore, we can represent a 'Plane' and its Position with a Part in the Mechanism-Editor and a Line in the Part-Editor.

What is the Design Statement of Position Synthesis?

Design Input:

Guide a Plane or Part through a number of specified positions.

Design Output:

A four-bar 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 Constraint-Based Sketch Tools

There is an example of each method in the Three-Position Synthesis.

Four and Five Position Synthesis:

Procedure 4: Uses more Advanced use of the Constraint-Based Sketch Tools

Rather than find a solution for cubic curves, we use the powerful Constraint-Based Sketch Tools in the Part-Editor 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.

Two Position Synthesis

To guide a Plane from position to a different position, at position

Representation of Two Positions for Synthesis

Step 1: We use the Part-Editor to draw the two Lines at the two positions.

Because each Line has an Origin and an X-axis direction, we can use it to represent a Plane.

The two positions are:

Position of Plane n	Plane Origin X(mm)	Plane Origin Y(mm)	Θ(degrees) of X-axis/Line/Plane
Position 1	-3	53	70
Position 2	31	59	30

Perpendicular Bisectors

Method 1

1.	Select two Points in the Plane at each position.

In this case, it is easy to select the start-Point and the end-Point of each Line.

2.	Draw a (red) construction line between the Points in each Plane at each position.

3.	Add a (green) construction line that is perpendicular to the mid-point of each red 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.

Intersection of Perpendicular Bisectors

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 centre of rotation for a Part that passes through Points and at the two Plane positions.

We need to transfer the learnings from the Part-Editor to add a mechanism in the Mechanism-Editor.

Measurements to get Link Lengths of Mechanism

Point can be the Pin-Joint of a Part that rotates.

1.	We can now make various measurements in the Mechanism-Editor 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 centre 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

Mechanism Link Rotation for Two Position Synthesis

You have all the information to:

1.	Add a Part

2.	Join it with a Pin-Joint and add a Motion-Dimension FB with the correct Base-Value - 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.

Three Position Synthesis

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 X-axis/Line/Plane
Position 1	-3	53	90
Position 2	31	59	60
Position 3	60	50	0

Procedure 1: Paper Method or Traditional Method

3Pos-1

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 Part-Editor

Sketch Lines as the three Positions of the Part in the Mechanism-Plane

1.	Edit the Base-Part

2.	Add three Lines

3.	Make the Lines the same length.

4.	Use constraints, dimensions or the Specify Point and Vectors dialog-box to fix the Part.

I have labelled the Points in the image as: A1,B1, A2,B2, A3,B1.

3Pos-P1-2

In the Part-Editor

1.	Draw two arcs, centred at Positions A1 and A2

Draw the Arcs so that they intersect at two places.

2.	Draw a Line through the intersection of the Arcs

This is a Perpendicular Bisector.

3.	Do steps 1 and 2 again, but for Positions A2 and A3

4.	Extend the two Perpendicular Bisectors until they intersect at Point A0

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.

3Pos-P1-3

1.	Do 1 to 4 again for Points B1,B2 and B3 to find Point B0 , and radius B0B

The construction is over.

You can draw arcs in the Part-Editor.

3Pos-P1-4

In the Mechanism-Editor

•	Measure, with Measurement FBs, the distance between A0 and A1, and also B0 and B1

•	Points, A0 and B0 are the fixed Pin-Joints in the four-bar mechanism.

3Pos-P1-5

•	Add two Parts, PART 1 and PART 2, with lengths A0A1 and B0B1, and join them to the Base-Part at A0 and B0.

•	Add a third Part with length AB.(this is defined as part of the specification)

•	Join the third Part to the free ends of the other two Parts.

•	We usually call Part 3 the 'coupler'

The four-bar mechanism (the Base-Part 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.

Procedure 2 - Basic Use of Constraint Based Tools - but thinking of Procedure 1

This procedure is only slight better than procedure 2.

It uses the same 'Perpendicular Bisector' ideas.

However, we introduce a few more Constraint Tools.

3Pos-1

In the Part-Editor

Sketch Lines as the three Positions for the Part in the Mechanism-Plane

1.	Edit the Base-Part

2.	Add three Lines

3.	Make the Lines the same length.

4.	Use constraints, dimensions or the Specify Point and Vectors dialog-box to fix the Part.

I have labelled the Points in the image as: A1,B1, A2,B2, A3,B1.

3Pos-2

Add the Geometric Construction to find the Centres and the Lengths of the Parts.

1.	Construct a perpendicular bi-sector between Points A1 and A2.

(Add a Line between Point A1 and A2. (A1A2)

Add a different Line. Constrain one end to the mid-Point of A1A2, and also make it Perpendicular to A1A2)

2.	Do Step 1 again, but for A2 and A3

3.	Add a Coincident Constraint between the end-points of the two perpendicular bisector. Call this Point A0.

4.	Do Steps 1, 2 and 3 again, but for Point B1,B2, and B3, to find Point B0.

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.

3Pos-3

In the Mechanism-Editor

1.	Measure the Distance between Point A0 and A1, and the B0 and B1.

Use Measurement FBs.- Note shown in image.

2.	Add Parts between Points A0 & A, Point B0 & B and A and B.

3.	Edit the lengths of the Parts to equal the Measurements.

(I have added Blue lines to show the four-bar mechanism).

3Pos-4

4.	Auto-Profiles and Auto-Layer tools.

5.	Add Motion-Dimension and other FBs to move the mechanism between the three positions.

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 infinity-squared number of choices.

Procedure 3: Advanced use of Constraint Based Tools

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 Mechanism-Editor to add the Parts with the correct lengths.

3Pos-1

This procedure is much easier and much quicker. It uses the Constraint Based Sketch Editor

In the Part-Editor

Sketch Lines as the three Positions for the Part/Plane in the Mechanism-Plane

1.	Edit the Base-Part

2.	Add three Lines

3.	Make the Lines the same length

4.	Use constraints, dimensions or the Point Vectors and Position dialog box to fix the Points.

You can label the Points: A1, B1, A2, B2, A3, B1.

3Pos-5

Add circles and Coincident Constraints

5.	Add a Circle anywhere in the Mechanism-Plane

6.	Make the circle coincident with A1, A2 and A3

7.	The Centre of the Circle is A0

8.	Add a different Circle

9.	Make the circle coincident with B1, B2 and B3.

10.	The Centre of this Circle is B0

3Pos-6

In the Mechanism-Editor

1.	Measure the Radius of the Circle that is coincident with A

2.	Measure the Radius of the Circle that is coincident with B.

Use Measurement FBs.- Note shown in image.

3.	Add Parts and Pin-Joint between Points A0 & A, Point B0 & B and A and B.

4.	Edit the lengths of the Parts to be equal to the Measurement given in 1 and 2

(I have added Blue Lines to show the four-part mechanism).

3Pos-7

5.	Use the Auto-Profiles and Auto-Layer tools.

6.	Add Motion-Dimension and other FBs to move the mechanism between the three positions.

Four Position Synthesis

4Pos1

In the Part-Editor

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 dialog-box.

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 X-direction.

I have labelled the Origins as 01, 02, 03, 04.

I have labelled the Points along the X-axis as X1, X2, X3, X4.

4Pos2

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 remain congruent.

4Pos-4a

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.

7.	Add a Circle

8.	Add Coincident Constraints between the Circle and the Apex of the four triangles, A1 to A4.

9.	Label the Centre of the Circle A0

Step 8 is complete.

4Pos5a

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 centres of the Circle B0

Step 12 is complete

4Pos7

Add Parts in the Mechanism-Editor

1.	Add a Part and Join it with a Pin-Joint to centre of a Circle

2.	Add a different Part and Join it with a Pin-Joint to the centre 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 Motion-Dimension FB to move it through the arc

4Pos4a

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:

•	different positions for the Pin-Joints in the frame

•	different lengths for the Parts

•	different positions for the Pin-Joints in the Coupler Part

You can experiment.

Different results may be better. For example the transmission angle of the Parts may be improved.

4pos4b

FIVE POSITION SYNTHESIS IS VERY SIMILAR.

YOU MUST MOVE THE CIRCLES UNTIL A FIFTH POINT ALIGNS WITH THE CIRCLES...

Five Position Synthesis - not complete, images only.

Position n	X(mm)	Y(mm)	Θ(degrees)
1	50	45	5
2	45	150	35
3	90	160	5
4	110	125	-45
5	65	110	-70

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

5pos3b

5pos4b

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.

Mechanism Synthesis: Five Position Synthesis-Coupler Points

Mechanism Synthesis: Five Position Synthesis-complete - circle point circles

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.

Mechanism Synthesis: Five Position Synthesis-complete construction

Mechanism Synthesis: Five Position Synthesis-complete

The Far image shows the same construction, but in the Mechanism-Editor.

The near images shows the four-bar 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 Part-Editor.

Five Position Synthesis - not complete, images only.

PS5-1

Position n	X(mm)	Y(mm)	Θ(degrees)
1	21.0	0.77	0
2	19.9	-4.53	-21
3	19.1	-11.94	-39
4	19.69	-17.71	-64
5	20.33	-27.57	-90

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

PS5-2

PS5-3

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.

PS5-4

PS5-6

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.

PS5-5

PS5-7

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 four-bar mechanism chain.

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