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Polynomial Cam Function (Fifth-degree polynomial Example) - Part 4

In the post [Polynomial Cam Function (Part 3)], we explored the characteristics of the 5th-degree profile. Now, we put it to work.

We will revisit the same die-press example from [Timing Diagram (Part 4)]. Originally, without overlap, the acceleration was a massive 4.154 m/s2. Using a Cycloid profile with overlap, we reduced it to 0.804 m/s2.

This time, we will use a Fifth-Degree (3-4-5) Polynomial combined with a Linear Cam Function. While the peak acceleration reduction might be similar to the Cycloid, the real advantage here is control—specifically, the ability to blend different motion segments seamlessly without stopping.

The Motion Strategy

Our constraints remain the same: Total displacement is 50 mm, and the die must dwell inside the hole for 30 mm.

Unit Conversion Note:
The equations we derived calculate velocity in mm/rad. To convert to real-world time domain:
• Velocity (mm/s) = Velocity (mm/rad) × Ï‰ (rad/s)
• Acceleration (mm/s2) = Acceleration (mm/rad2) × Ï‰2

The Displacement Profile

We have divided the motion into specific sectors to optimize the flow. Notice how we use a Linear segment (A-B) to handle the approach, connected to Polynomial segments for the heavy lifting.

Sector Function Start Angle (°) End Angle (°) Start Height (mm) End Height (mm)
A-B Linear 62 77 1.43 0.43
B-C Fifth-Degree 77 206 0.43 50.00
C-D Dwell 206 306 50.00 50.00
D-A Fifth-Degree 306 62 50.00 1.43

Velocity Matching: The Secret to Smoothness

This is where the polynomial shines. To connect the Linear segment (A-B) to the Polynomial segment (B-C) without a shock, the velocities must match perfectly.

Calculating the Slope:
Slope (Velocity) = (0.43 - 1.43) / (77 - 62) = -0.0667 mm/deg

We input this value (-0.0667) as the Start Velocity (v0) for our polynomial calculation in sector B-C. This ensures a perfectly continuous transition with zero jerk spike.

Acceleration Analysis

Using motion simulation in Excel, we can visualize the result. We verify that there are no infinite spikes (Jerk) at the connecting points.

Modern Application: Electronic Camming (E-Cam)

Today, physical cams are being replaced by Electronic Camming (E-Cam) profiles inside servo drives. However, the math remains identical.

By programming these 5th-degree polynomials into your servo controller, you can achieve smoother motion, higher speeds, and longer machine life—all without changing a single mechanical part.

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