Thank you for your complimentary remarks. My purpose was to introduce readers to polynomial applications and act as a primer for the techniques required. We live in a "point and click" world, and even in engineering, it seems that tasks are done with the least possible effort and thought. I hope I’ve provoked some real thought on the development of servo system motions.

I focused on the 3–4–5 polynomial in light of my presentation’s introductory nature. In my nearly 40 years of working with a variety of motion applications, I’ve enjoyed great success with the 3–4–5. The real power of polynomial techniques is the design of more specialized requirements.

Regarding the 4–5–6–7 polynomial, the zero-jerk specification at the beginning and end of the motion is an advantage for vibration characteristics when compared with the 3–4–5. However, the 4–5–6–7 produces extremely small displacement at the beginning and end of the motion. For mechanical cams, the accurate fabrication of such small displacements remains a problem. In fact, this is an issue, albeit to a lesser extent, for all motion curves that have zero acceleration at the motion terminals. Depending on a servo system’s resolution, accurately generating the 4–5–6–7 is perhaps more attainable.

Recognizing that the first and last portions of the 4–5–6–7 displacement are virtual dwells, the motion’s period should be increased to reduce acceleration and velocity. If the 4–5–6–7 is arbitrarily substituted for a 3–4–5 using the same time period, acceleration will increase by 30%, and velocity will increase by 17%. To have an acceleration equal to the 3–4–5 for a given time period, the 4–5–6–7’s time must be increased about 14%. To have equal velocities, the 4–5–6–7’s period must be increased by about 17%. This illustrates the need to understand the characteristics of the various motion forms and the impact that a designer’s decisions have on the device’s resulting performance.