3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Introduction

In our previous tutorials, such as [3-Position Motion Generation Synthesis with Alternate Moving Pivots], we used a "standard" synthesis approach. We defined the moving coupler first, and the geometry dictated where the ground pivots (O₂ and O₄) had to be.

But what if you don't have that freedom?

In real-world machine design, you often have a pre-existing frame or base. You cannot drill holes just anywhere; the ground pivots must be located at specific, available points. In this scenario, the standard method fails because it gives you valid kinematic solutions that might require mounting a pivot in thin air or inside a motor.

The Solution: Kinematic Inversion

To solve this, we use the Inversion Method.

The Core Concept:
Instead of looking at the mechanism from the perspective of a stationary ground and a moving coupler, we invert our perspective. We pretend the Coupler is stationary and the Ground is moving.

By fixing the coupler positions (A₁B₁, A₂B₂, A₃B₃) on top of each other, we can rotate the ground pivots backwards relative to the coupler. This allows us to geometrically find the required length of the input and output links that connect our specific ground points to the moving coupler.

Why use the Inversion Method?

• Fixed Ground Constraints: Essential when you must attach links to specific pre-drilled holes or mounting pads on a machine frame.
• Compact Design: Helps in designing mechanisms that must fit within a tight housing where pivot locations are limited.
• Complex Motion: Often easier to visualize relative motion between the end-effector and the base.

What to Expect in Part 2

In the next post [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2], we will walk through the step-by-step CAD procedure:

1. Defining the 3 desired positions of the coupler.
2. Defining the Fixed Pivot Locations (O₂ and O₄) first.
3. Using CAD transformations to "invert" the ground points.
4. Finding the center of the circle defined by these inverted points to determine link lengths.

This method is a powerful tool in the arsenal of any Mechanism Design Engineer.

Recommended Reading

• Kinematic Synthesis of Linkages (Hartenberg & Denavit) - The classic text on inversion.
• Design of Machinery (Norton) - Excellent visual examples of inversion.