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In the previous introduction, we established the problem: We have fixed mounting points (O2 and O4) on our machine base, and we need to design a linkage to hit 3 specific positions.
Standard synthesis moves the pivots to fit the motion. In Kinematic Inversion, we do the opposite: we virtually move the ground to fit the coupler. By "freezing" the coupler in Position 1 and moving the ground relative to it, we can geometrically find the required link lengths.
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Step 1: Setup the Constraints
Start by drawing your known constraints in the CAD Sketcher (NX, SolidWorks, etc.):
1. The Fixed Ground Pivots (O2 and O4).
2. The 3 Desired Coupler Positions (A1B1, A2B2, A3B3).
Figure 1: The setup showing fixed grounds (bottom circles) and the target motion path (red lines).
Step 2: Inverting Ground Pivot O2
Now we perform the "Inversion." We need to find where the ground pivot O2 would be relative to Position 1 if the coupler stayed still.
Finding O'2 (Relative Position 2):
Measure the geometric relationship (distance and angle) between ground O2 and coupler A2B2. Recreate this exact relationship attached to A1B1.
CAD Tip: Define a rigid triangle O2-A2-B2, copy it, and align the A-B side to A1-B1. The new location of O2 is your inverted point O'2.
Measure the geometric relationship (distance and angle) between ground O2 and coupler A2B2. Recreate this exact relationship attached to A1B1.
CAD Tip: Define a rigid triangle O2-A2-B2, copy it, and align the A-B side to A1-B1. The new location of O2 is your inverted point O'2.
Figure 2: Defining the rigid triangle O2-A2-B2.
Figure 3: Inverting the ground pivot to find point O'2 relative to the first position.
Finding O''2 (Relative Position 3):
Repeat the process for Position 3. Map the relationship of O2 relative to A3B3 back to A1B1.
Figure 4: Capturing the geometry for the third position.
Figure 5: We now have 3 relative ground points: O2, O'2, and O''2.
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Step 3: Finding the Moving Pivots (G and H)
We now have three points (O2, O'2, O''2) that represent the path of the ground relative to the coupler. To find the fixed pivot on the coupler (the "Moving Pivot"), we find the center of the circle described by these three points.
Finding Moving Pivot G:
Draw chords between O2-O'2 and O'2-O''2. Construct perpendicular bisectors. The intersection point G is the moving pivot on the coupler.
Result: Link 2 is defined as the line connecting the real ground O2 to G.
Figure 6: Intersection of the perpendicular bisectors locates Moving Pivot G.
Finding Moving Pivot H:
Repeat the entire inversion process for ground pivot O4. Find relative points O'4 and O''4. Bisect the chords to find intersection point H.
Result: Link 4 is defined as the line connecting the real ground O4 to H.
Figure 7: Locating the second Moving Pivot H.
Step 4: The Final Linkage
Connect your real grounds to your new moving pivots:
1. Input Link = O2-G
2. Output Link = O4-H
3. Coupler = G-H-A1-B1 (Rigidly connected)
Figure 8: The completed linkage mechanism.
Let's verify this in the Part 3 video simulation.
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