Three-Position Synthesis

Three-position synthesis is used to design a four-bar linkage that moves a coupler through three specified positions. There are two common cases: Finding Fixed Pivots and With Fixed Pivots.

Finding Fixed Pivots

In this case, you determine the fixed pivot locations based on the three coupler positions.

1. Take a screenshot from the reference video to define the three coupler positions
2. Draw a Segment with points P1 and P2 on top of the first coupler position
3. Use Compass with P1 and P2 as radius
4. Place the compass center at the first point of the second position (P3)
5. Draw a Segment from P3 to P4, where P4 is on the compass circle and aligned with the second position
6. (Optional) Hide the compass circle
7. Use Compass again with P1 and P2 as radius
8. Place the compass center at the first point of the third position (P5)
9. Draw a Segment from P5 to P6, where P6 is on the compass circle and aligned with the third position
10. (Optional) Hide the compass circle

11. Draw a Segment between P1 and P3
12. Draw a Segment between P3 and P5
13. Construct the Perpendicular Bisector of segment P1-P3
14. Construct the Perpendicular Bisector of segment P3-P5
15. Place a point P7 at the intersection of the two bisectors as the first fixed pivot

16. Repeat steps 11-15 for P2, P4, and P6 to find the second fixed pivot (P8)

17. Sketch the linkage on top of the geometry construction

Discussion

The above example yields a double-rocker linkage with limited angular motion bounded by toggle positions. The mechanism cannot fully reach the third position due to its kinematic limits — a key distinction between synthesis intent and feasible motion in non-Grashof mechanisms.

Actuation choice also matters: driving the other rocker causes the coupler to follow a different motion branch, failing to reach positions 2 and 3 due to kinematic singularities. This input-output asymmetry means driving different rockers is not equivalent.

For limited-motion mechanisms, reachability and actuation must be evaluated after synthesis, especially near toggle configurations.

With Fixed Pivots

In this case, you are given the fixed pivot locations and need to find the moving pivots on the coupler.

1. Take a screenshot from the reference video to define the three coupler positions and fixed pivots
2. Draw a Segment with points P1 and P2 on top of the first coupler position
3. Use Compass with P1 and P2 as radius
4. Place the compass center at the first point of the second position (P3)
5. Draw a Segment from P3 to P4, where P4 is on the compass circle and aligned with the second position
6. (Optional) Hide the compass circle
7. Use Compass again with P1 and P2 as radius
8. Place the compass center at the first point of the third position (P5)
9. Draw a Segment from P5 to P6, where P6 is on the compass circle and aligned with the third position
10. (Optional) Hide the compass circle
11. Draw a Segment with points P7 and P8 on top of the given fixed pivots

12. Use Compass with P3 and P7 as radius, P1 as center
13. Use Compass with P4 and P7 as radius, P2 as center
14. Place a point P9 at the intersection of the two compass circles
15. Use Compass with P3 and P8 as radius, P1 as center
16. Use Compass with P4 and P8 as radius, P2 as center
17. Place a point P10 at the intersection of the two compass circles
18. (Recommended) Hide the compass circles
19. Draw a Segment between P9 and P10

tip

The pattern of creating two compass circles and finding their intersection is used to copy a triangle when one side has the same length. For example, triangle P3-P4-P7 is copied to P1-P2-P9.

20. Use Compass with P5 and P7 as radius, P1 as center
21. Use Compass with P6 and P7 as radius, P2 as center
22. Place a point P11 at the intersection of the two compass circles
23. Use Compass with P5 and P8 as radius, P1 as center
24. Use Compass with P6 and P8 as radius, P2 as center
25. Place a point P12 at the intersection of the two compass circles
26. (Recommended) Hide the compass circles
27. Draw a Segment between P11 and P12

28. Draw a Segment between P7 and P9
29. Draw a Segment between P9 and P11
30. Construct the Perpendicular Bisector of each segment
31. Place a point P13 at the intersection of the two bisectors as the first moving pivot
32. Repeat steps 28-31 for P8, P10, and P12 to find the second moving pivot (P14)

33. Temporarily treat P13 and P14 as fixed pivots and P7 and P8 as moving pivots
34. Sketch a four-bar linkage
35. Add an actuator to the joint at P14
36. Verify the four-bar passes through segments P7-P8, P9-P10, and P11-P12

37. Remove the actuator
38. Invert the four-bar linkage by making the link P7-P8 the new ground
39. This inverts P13-P14 back to moving pivots and P7-P8 to fixed pivots
40. Extend the coupler (now the link P13-P14) to include the original segment points P1 and P2
41. Add an actuator at the joint of point P8