For engineers who already know the math—but still lose projects. For the last few years, I’ve been sharing technical guides here on Mechanical Design Handbook —how to size a motor, how to calculate fits, and (as you recently read) how to choose between timing belts and ball screws. But after 25 years in industrial automation, I realized something uncomfortable: Projects rarely fail because the math was wrong. They fail because: The client changed the scope three times in one week. A critical vendor lied about a shipping date (and no one verified it). The installation technician couldn’t fit a wrench into the gap we designed. University taught us the physics. It didn’t teach us the reality. That gap is why I wrote my new book, The Sheet Mechanic . This is not a textbook. It is a field manual for the messy, political, and chaotic space between the CAD model and the factory floor. It captures the systems I’ve used to survive industrial projec...
Figure 1: A modern linear ball slide (like this THK model) is the contemporary solution for precise straight-line motion.
Many modern engineering applications require components to move in a precise linear fashion, known as "straight-line motion". Today, we take this for granted. We can simply purchase an off-the-shelf Linear Motion Guide that moves a device accurately along a rail with low friction.
The Historical Challenge: Making a Straight Line
However, in the late 17th and early 18th centuries—before the development of high-precision milling machines—it was extremely difficult to manufacture long, perfectly flat surfaces. Creating a sliding joint without significant backlash was nearly impossible.
During that era, engineers had to rely on Linkages. Much thought was given to the problem of attaining a straight-line motion using only revolute (hinge) connections, which were much easier to manufacture.
The most famous early result was the mechanism developed by James Watt for guiding steam engine pistons. While revolutionary, Watt's linkage did not generate an exact straight line; it produced a figure-eight curve with a straight-line approximation over a limited distance.
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Enter the Roberts Mechanism
In this post, we explore a more refined approximated straight-line mechanism discovered by the Welsh engineer Richard Roberts (1789–1864).
Figure 2: The Roberts Mechanism is a symmetric four-bar linkage with a rigid triangular coupler extending down to point C.
The Roberts Mechanism is a symmetric four-bar linkage. It is distinct because the coupler (the triangle in the middle) is a rigid body that extends downwards to a tracing point.
Kinematic Geometry
For the mechanism to function correctly and trace an approximate straight line at point C, specific geometric ratios must be maintained. It requires the linkage to be symmetric.
Specific Lengths for Straight Line Generation
- O2A = 100 (Crank)
- O4B = 100 (Rocker)
- AB = 100 (Top of Coupler)
- AC = 100 (Side of Coupler)
- BC = 100 (Side of Coupler)
- O2O4 = 200 (Ground Distance)
Rule of Thumb: The ground distance should be twice the length of the ground-pivoted links, and the coupler should be an isosceles (or equilateral) triangle.
Simulation and Motion Analysis
We can observe from the simulation below that point C moves in an approximate straight line between the two ground pivots.
It is important to note that this is not exact. If you plot the full path of point C, it traces a curve resembling a "W" with a flat bottom. However, for the central portion of the travel, the deviation from linear is minimal.
Video 1: Kinematic simulation showing the straight-line motion of point C.
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Modern Applications: Flexures
While originally designed for steam engines and textile machinery, Roberts' mechanism has found new life in modern high-precision engineering, specifically in Compliant Mechanisms and Flexures.
Figure 3: A visualization of the path. The central portion is highly linear, useful for precision applications.
By replacing the pin joints with flexible flexure hinges, engineers can create straight-line motion stages with unique properties:
- No Friction: Because it relies on bending rather than sliding or rolling, it generates zero friction.
- No Lubrication: Perfect for vacuum, clean-room, or space environments.
- High Precision: Eliminates the "stick-slip" phenomenon found in mechanical slides.
Designing with Software
Modern design and analysis of such linkages are typically done using software like SAM (Synthesis and Analysis of Mechanisms) by Artas Engineering or other CAD packages with kinematic modules. These tools allow you to modify link lengths on the fly and immediately visualize the coupler path to optimize straightness for your specific stroke.
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